Mozzie bites do that to me too. And they stay for days. I've got scars. Needless to say, living in Louisiana, I keep a supply of Benedryl (pill and cream form) handy at all times.As others have said, sounds like an allergic reaction.
I have an allergy to mosquito bites. When I get bitten by a mosquito the bites swell up to 3 or 4 times the size of a normal bite. Used to be worse when I was a kid. I got a bite on my face when I was 5 or 6 that swelled the skin under my eye to the size of a golf ball
Benadryl, Calamine Lotion, and Witch Hazel used to work pretty well for me.