"in order to 'better' solve a problem" sounds uncommon, is something wrong?

“better” is being used as an adverb here. You can also do the same with “best”.

If X is a better (or the best) solution to problem Y, then you can say that “X better (or best) solves Y”.

Note that X is often implicitly compared to the status quo, such as “we can better serve our customers by staying open an hour later (than we do today).”

The specific structure “to [adverb] [verb]” is called a “split infinitive”, and some purists dislike it, but it is nevertheless commonly used and accepted. The canonical example is “to boldly go” from Star Trek.

"better" is also a verb:
His total of five gold medals is unlikely to be bettered.

You can also find the similar structure in the book of "Interdisciplinarity for the 21st Century" page 113, ( in order to better solve it)