Sharepoint online - Change position of links and picture in header

Here is the solution of your problem. First you need Modern Script Editor

Once this is done then you need to change this CSS.

.topNavBox-62 {
position: absolute;
top: 0px;
left: 0px;
right: 0px;
line-height: 22px;
padding-left: 32px;
padding-right: 32px;
}

To this,

.topNavBox-62 {
position: absolute;
top: 6em;
left: 0px;
right: 0px;
line-height: 22px;
padding-left: 32px;
padding-right: 32px;
}

Hope this help you.