"Can Fanfiction Make You a Better Writer?"
Yes, but only while writing it. The best things I've ever written were fan-fiction of Lord of the Rings and Diablo. That's because I already had an established medium, with set limits, stages, characters, continuity. Everything was there, all I had to do was weave them in a way I liked.
When writing original work, I always fall in to the vicious circle of reiterating. I don't have a style set in stone, a limit, or actually a perfect ending. So I feel compelled to rewrite, improving the quality, the detail, but delaying to the point where I've written hundreds of pages in 5 years, all of them being rewrites of the same 30 pages I started with years ago.
Fan-fiction is a lot easier.