I wanted to write: I understood some of it. So I thought Bazi anladım. But I was told it should be biraz anladım.

In English, that would be like understanding 10-25%. But I understood about 30-40%. So, either, is there a word in Turkish that is more than biraz but less than çok? Or could biraz mean more than ´a little´ in the English sense?

"Bazı" is an adjective and not used as an adverb. It´s pretty close to the English word "certain" as in "certain people" (~"bazı kişiler").