Renormalization and yet missing mathematical proofs?

I learned that renormalization used in the standard model while working is using some math that has not yet been proved. Put very simple renormalization is subtracting an infinite number from another infinite number to get a finite number. I learned during a lecture on quantum field theory that there is no mathematical proof yet for some of the mathematics used in renormalization of the standard model. However the experiment show it works. So it’s sort of math empirically confirmed?

My question now is what are the exact things with renormalization that are not yet proved mathematically? Could you guide me towards some literature/papers talking about this?

I found some questions along the way like this and this but they seem not to provide an answer to my question.