It is normal, and to be expected in certain situations. The trick is avoiding those situations, since most people would prefer to avoid being bitten (and I recommend it).
But lets suppose you just got bit. You don't punish the bird. That simply won't be understood and will lead to far worse problems. Walk away and let your own emotions cool down. Then ask yourself why the bird bit you. It may not know you yet and was frightened by whatever you were doing. It may have been asleep, napping with its eyes open and you surprised it by sticking your hand in the cage. There can be all kinds of explanations.
Most birds will warn you, but you do have to learn their body language. Most birds will try to scare you away or bluff. Some birds are just excited and don't know what to do with themselves. Some birds will bite you because you are handy and someone they want to bite is not readily accessible. Some birds have moods. They may try to bite you now, but if you go away and come back in a few minutes they will be happy to get up on your arm or whatever.
I did have one bird that I always had to use a stick to get out of his cage. Then once I got him out and aways from other birds, he became the nicest of all birds. You have to view all of this as a puzzle to solve and sort out.
The important thing is not to retaliate, and not to let resentment (or even fear) take root in your own heart.
I am pretty sure books could be written about all of this. You can also learn things about human behavior (as well as about yourself) by working with birds.
I should note in closing, that some birds never bite. But I thoroughly enjoy some of my "difficult birds" that have some attitude and spirit.
Tom's parrot pages / [email protected]