Too equates to more than necessary, which isn't what you are wanting to say, apparently. So equates to very.
Notice that "I don't believe I make too many mistakes in Turkish" is perfectly normal; and "I don't believe I make so many mistakes in Turkish" sounds a bit odd.
So is a strengthening intensifier, and wants consistency from can't, which is also a strenghtened version of don't. Awkwardly, words that weaken the situation are also called intensifiers. Your sentence is trying to strengthen quantification of the mistakes, so can't and so work here. My sentence with don't is trying to weaken the quantification of the mistakes, so don't and too work as a pair.
Looked for, but didn't find, collocation data and analysis for verbs do, can and intensifiers too, so.