There are a few different things that could be the problem, but either way, no, that isn't quite normal. That doesn't mean that you're a freak or that there is necessarily anything medically wrong with you either though. It just means that something is not quite right, probably something very small and easy to fix.
The vagina is made up of some very sensitive tissue. Fingers are kind of rough. Fingernails can cut the skin. Also, many men are far too rough with their hands. So take a look at his hands first. See if his nails need to be cut. If his hands are very rough, you could always have him put a condom over his finger or fingers. It might sound silly, but condoms are smooth and often coated with lubrication.
Another problem could be lubrication. Lack of lubrication is the number one cause of pain during sex. If you weren't lubricated enough, that makes your vagina even more likely to get hurt. If you are often dry, you can buy more lubrication at the store; water-based is usually best, so read the label.
If something that he is doing is hurting you, tell him right away. Don't let him continue to do it that way. There is no reason why you two cant continue what you were doing after asking him to be more gentle. Sexual activity sometimes requires a lot of communication.
Now, there is a small chance that something a little more serious is going on down there. There is no harm in seeing an ob-gyn just in case. It could even be a cut that isn't healing and keeps opening up. She might be able to just give you a cream. It's kind of embarrassing to go to the ob-gyn, but it feels great to come out of there with your questions answered. We can't really diagnose you over the internet, and there are too many possibilities, so we will probably of limited help. Just don't go to the ob-gyn when you have your period.