Is it normal to have a 13 year old girlfriend?

If you are around that age, yes, according to what I've been seeing around. As in, they like each other and everyone knows it.

But I would strongly suggest that you not, because relationships at this age rarely ever last past college to marriage. If the two have sex too early, the female can get pregnant and if she gives birth to that baby there will be consequences that will affect the rest of her life...and possibly the baby.

So if you want to have sex, make sure you don't get her pregnant too early. She may not be ready - wait a few years. Maybe by then you'll find someone else who was truly meant for you, if she wasn't.