Why should browser security be prioritized?

• Most of the websites I visit have SSL certificate,
• Most of them are either big enough that I can trust that they can't be hacked, or small enough that I don't think it's profitable for the hackers
• I can smell fishy websites?

Everything in there is wrong.

Browser security, as you guessed, is important because on most systems, no other piece of software so regularily encounters data from outside the trust boundary that

• needs to be processed in near realtime,
• does not subscribe to some simple and easy to filter format and
• is then parsed or executed by a highly complex software that implements a constantly changing target - or in other words: Is highly likely to contain implementation errors, some of which may be exploitable.

SSL protects you against certain types of attacks, most of which are of no relevance if the attackers goal is to compromise your machine. As its more current name - TLS - implies, its business is transport security, not protecting your browser or your machine.

Websites of all types get hacked constantly. The smallest ones by automated bots. The background noise here is quite astonishing. If you have your own server, turn up your logfiles for a day and marvel at the sheer amount of connection attempts to pretty much all interesting ports, 24/7. Turn up your webserver logfiles and read through the strange paths and standard routes to popular, insecure software packages. This stuff would not be out there if it would not work, at least sometimes.

The large websites, meanwhile, are not more secure just by being bigger. In the animal kingdom, you have to be the biggest to be safe from predators. If you are just big, it just means you are being hunted by lions instead of foxes. Same on the Internet, being a large website just means your threat actors are dedicated activists, organised crime, maybe nation states (depending on your business), etc. The threat profile changes based on your size, but it is utterly naive to state outright that it changes for the better.

You can probably smell fishy websites, but quite a bit of malware has been deposited through sideloading, e.g. through an advertisement network instead of the websites directly. It could be through a compromised CDN that your utterly non-smelly website uses.

So, in summary: Browser security matters because it is one of the largest if not the largest hole in your perimeter, and you can not protect yourself adequately through browsing habits or anti-malware tools, though both do help.

Since nobody pointed this out yet:

Anti virus software is way less useful than you think. In fact, if you ask security professionals, the main recommendation to stay safe will be to update all your software, above the recommendation to run an AntiVirus - so exactly what you're planning not to do.

Why? All that anti virus software does is stop known, old pandemics. However, most successful widespread attacks simply create new viruses and use the time window until anti virus software is updated to spread. Anti virus software is no magic bullet, it only detects what it knows (there is behavior detection but it isn't very effective for any of the AV software out there) so against new attacks, it only helps to have less intrusion opportunities. And by far the cheapest intrusion opportunity for attackers is an outdated browser or core system.

So yes, it is vital that your browser is up-to-date if you care about any of the data on your computer at all. Whether you have an Anti Virus is actually kind of unimportant - it does help a little, but it won't save you against most new attacks if your browser or core system are outdated.

Side note: this might be different if you run a browser sandboxed. However, unless you know that is what you're doing, you're most likely not doing it. And I'm not sure for 32bit, any of the common sandboxing solutions like UWP/Windows App Store is supported at all. Also, in that case websites might still be able to steal data from all other open websites, including your banking website tab. So that's hardly ideal.

Browser updates include updates to certificate authority data and indeed its very own certificate. Both of those things have expiration dates and there are many websites which can't be accessed at all by too-old browsers because the SSL certificate can't be verified by that too-old browser OR that too-old browser can't verify its own cert to the remote server. So even if you don't care about security, you might care about the access and functionality which security-validation achieves for you.

What's more, by the time you "smell" the fishiness of a website you're looking at, it's already too late for many kinds of attacks. Hopefully your old 32-bit operating system isn't also unable to get updates to its antivirus software.

You should care about security if you care about security.

If you care enough to say things like "I can smell fishy websites" or "Most of the websites I visit have SSL certificates," then you do care about security.

Attacks are out there. Period. Let's not fool ourselves. Let's not pretend we're immune. Take Stuxnet. Stuxnet is credited with destroying up to 1000 centrifuges used for enriching nuclear fuel in Iran. That's hundreds of millions of dollars in damage, at the very least. Practically speaking, a good risk assessment from the country's perspective would be even more dire.

You feel invincible because you visit sites with SSL certificates? The computers Stuxnet hit were air gapped. There was no wire going between the internet and the computers that got infected. They still got infected.

So the real question is not "am I safe?" You're not. The real question is are you safe enough. Is it worth spending some number of dollars and some number of hours to upgrade? That's a balance question. That's much more useful.

Here's a test. Log into your computer. Log into a few sites you care about. Now hand me your computer. I'm going to walk away with it. How do you feel? How nervous are you that I have your data? If all you have is a harddrive full of cute kitten pictures, then you are probably going to be more angry that I walked off with your laptop. Laptops are expensive to replace. But if you've got your banking credentials on that computer, you're probably going to be more worried about what I can do with those. Is your SSN on that computer?

Once you understand how much damage can be done, versus the cost of preventing the damage, you can make a judgement call.

Each of your statements is making a false assumption here:

Most of the websites I visit have SSL certificate.

This is great, but SSL/TLS only protects you against certain types of attacks.

Pretty much, a site having a (valid) TLS certificate simply means that the owner of that website has in some way proven ownership of the domain name that is used to access the site, possibly spending a very large amount of money to do so (or possibly spending none). Usually, this means that you can trust that the site is who they say they are, barring domain spoofing (which I will cover in response to your third point). However, it may not mean that (see for example the fiasco that got everything issued by Symantec revoked/untrusted earlier this year). So, a TLS certificate really just verifies site ownership, not that the site is legitimate or that they are doing what they say they are.

TLS certificates give one more protection benefit, they let you use HTTPS. HTTPS provides exactly two types of protection if it's used correctly:

• It provides a reasonably high level of confidence that the data you are receiving is the same data that the web server sent. This is important for ensuring that nothing has been added to or removed from the site by a third party while it was in transit to you.
• It provides a reasonably high level of confidence that the data you are transferring is not visible in-transit to third-parties. This is the reason it's so important to ensure that any site you provide a password or personal data to uses HTTPS.

This still leaves you open to quite a few attacks. The most obvious is for the web server to be hacked (or the CDN if they're using one). There are others too, like XSS attacks, malicious advertisements (think of all the perfectly legitimate sites out there that have hidden auto-redirects to malicious sites because of the ads they use), and attacks on TLS itself (such attacks are why no sensible operator still uses SSLv2 or SSLv3, both are known broken). So, HTTPS/TLS/SSL by itself is not rigorous protection.

Also, as you yourself said, most of the websites you visit use TLS, not all of them. Think very hard about whether you really should trust those other sites that aren't using it, if they aren't willing to take the five minutes to set their servers up with free certificates from LetsEncrypt, what other aspects of security are they cutting corners on?

Most of them are either big enough that I can trust that they can't be hacked, or small enough that I don't think it's profitable for the hackers.

Have you not watched any news for the past decade? Just in terms of public disclosures, there are dozens of known attacks on sites with well over 100 million users (that's a big site by most people's definitions, as 100 million is more than 1% of the world population, and probably closer to 3-4% of the total number of people on the internet). I suggest taking a look at the list of public breaches on Have I Been Pwned?, there are quite a few big names on there, including ones which handle very sensitive data (Experian for example). So, no, it's not realistic to expect something to be 'too big to be hacked'. In fact, big sites are some of the most attractive targets for attackers, because they have lots of users. They also have a notoriously bad track record for publicly disclosing security breaches (they have more incentive not to, as they have more potential customers to lose).

On the other side of things, small sites are easy targets, even if not attractive. If you think of big sites as high risk investments for hackers, small ones are low-risk investments. They may not give as much in the way of returns, but they often will give much more consistent returns, so they're still attractive targets.

Also, consider that quite often attackers are targeting software that is vulnerable, not necessarily specific sites. WordPress is a good example, as it's used on sites both big and small, and it's been successfully used multiple times in the past as an attack vector.

I can smell fishy websites

First off, just because a site isn't 'fishy' doesn't mean it's not a threat. There are also quite a few legitimate sites that look 'fishy' by many people's definitions.

Second, it's not hard to copy a legitimate site but still do illegitimate things with the result. Domain spoofing, in it's various forms, is often used for this. There was a rather good blog post on a big infosec site a while back (which i unfortunately cannot find right now, otherwise I would link it here) demonstrating this with apple.com.

As an example of the type of thing you need to look out for but probably can't spot, can you tell the difference between uv and υν? No, this isn't a trick question, the first one is the lower case Latin letters 'u' and 'v', while the second one is the lowercase Greek letters upsilon and nu. In most sans-serif fonts (like those used by almost all browsers in the address bar, and the default font on most SE sub-sites), those two pairs of characters are nearly indistinguishable. Even in many serif fonts, they're hard to distinguish for most people. Similarly, АВ is actually a pair of Cyrillic characters, not Latin ones, and again is indistinguishable from the Latin 'AB' in most fonts. Both cases illustrate examples of IDN homograph attacks, a technique whereby attackers take advantage of the similarity of different characters looking similar or identical to trick people into following their links by making them look like links to legitimate websites.

Pretty much, don't assume that you will be able to recognize a site that's a threat until it's too late.

As an addition to the excellent points in other answers:

Is your 32-bit software still regularly updated or is it a deprecated old version?

Once any piece of internet connected software stops receiving new security fixes, it becomes susceptible to every new bug or hack thereafter.

Not all the websites you visit have certificates. You can’t smell fishy websites. Certificate doesn’t mean the site isn’t trying to hack you.

The browser is the biggest attack vector against your computer. It will tend to run unvetted JavaScript code at least, and god knows what else. It constantly processes data from untrusted sources.

Can you explain why browser security should be placed on the top priority ...

Because the browser is processing lots of untrusted content from the internet.

Of course, if you use any other programs which does this (like Mail client, maybe Office program, PDF reader) you should keep these updated too since vulnerabilities in these programs are a regular attack vector too.

.. Most of the websites I visit have SSL certificate,

A SSL certificate says nothing about the trust you can have in a site. HTTPS only protects against sniffing modification of the traffic during transport. A HTTPS site can serve malware as much as a plain HTTP site can do.

Apart from that "Most of the websites" is not the same as "All of the websites".

I can smell fishy websites?

Even if you might be confident in your ability to sniff websites where the URL looks fishy (which might actually be overconfidence) I'm pretty sure that you will not know up-front if the site you visit regularly got hacked and is serving malware (i.e. Watering hole attack or other kinds of hacking high-reputation sites to increase number of victims) or if it is serving malicious ads which are outside the control of the website itself (i.e. Malvertising).

EDIT:
After I've wrote my answer the OP added the following to the question:

Most of them are either big enough that I can trust that they can't be hacked, ...

Too big to be hacked? While large web sites usually employ better security than smaller ones it does not mean they are unhackable. And sites with lots of customers are especially a lucrative target for the attackers since this also means lots of potential victims. Some examples: ... malicious ads on Forbes ... or ... New York Times and BBC hit by 'ransomware' malvertising or Study: One-third of top websites vulnerable or hacked.

... or small enough that I don't think it's profitable for the hackers, ...

Too small to be hacked? That's not true either: attackers use automated tools to hack insecure CMS installations like WordPress or Django en mass, i.e. it is very cheap to take over a vulnerable site this way.