My girlfriend wants to use git to track progress on her artpieces, and doesn't want that to go into a public repo. Are there any remotes that explicitly support this, or should I just point her to BitBucket?
It's not always necessary that you store only source code in your git repositories; you can store anything, like PDFs, docs, images, etc.
As far as her artwork is concerned, I know it is important to keep them private. There are many other git providers out there offering free private repositories. Some of them are:
All of them provide free repositories. I would recommend (if not going for Bitbucket) Gitlab is also worth a try.