NO it is NOT normal.

This is happening because you are hitting the bag incorrectly ad it is dangerous, Also you may need to improve you method of putting on hand wraps and get some more padded - perhaps 10 or 12 ounce gloves.

Don't just keep boxing through it and if possible get some coaching. Even coaching at a kickboxing gym is likely to be better than nothing. You sometimes find boxing coaches operate out of the same places anyway.