Is it normal for me to bite myself to stop the urge to self harm?

Cutting is more dangerous/permanent than biting (that being said, I'm not aware how much pressure you're applying when you do.) So I suppose biting is a better alternative, though I use the term loosely. You should consider some help from family, friends, or a professional in order to stop self-harming entirely. It seems that you have some unresolved issues that cause you to do these things, possibly some mental disorder as well, however it's not my place to make those judgements. Instead of cutting you could try squeezing ice cubes, snapping a rubber band, etc..