Or "I understand much better now." If someone helped me with a translation and I responded with: どうもありがとうございます。前より上手に分かっています。 Would that be the correct way to express that in Japanese?
A typical phrase in japanese for this situation is "勉強になりました", though it is obviously not a direct translation. Maybe you could say:
Or from Chocolateさん:
I doubt one says "上手に"+"分かる", since 分かる is "come to be understood", rather than "actively understand".
The suggested 勉強になりました is pretty common.
If you really want to say "I understand much better than before", you could say "やっと、分かってきた", "やっと理解するようになった", or "前より理解できた". Although I would probably add "かも(しれない)" at the end of all these examples, to be somewhat less assertive about my comprehension faculties.