You were misinformed. That's exactly what it means. Clue: a lot = very much. To be clear, Thanks a lot = Thank you very much. It has no other meaning except that.

Thank you so much,Joshua C

"Thanks a lot" and "Thank you very much" mean pretty much the same thing. However, I would say that "Thank you very much" would sound a little more polite because I use "thanks a lot" mostly with my friends and people I know well only. Also, I think that "It's okay, don't bother/worry about me" is a little closer to the expression "please, you don't need to do this"