My son was the same way for a few months. Every bite he got blew up into a hard, red lump about two inches in diameter. His eye swelled shut once, one time he couldn't barely bend his arm because the bite was inside the elbow, his ankle looked like he had elephantitis, and sometimes it looked like he had goose eggs on his head! Now, months later, his body is calming down the reactions to them and they don't look so bad, but they ooze a sticky clear fluid.
We just ice them as he tolerates it, keep them clean and keep him from scratching them.
My sister reacted the same way to bites as a little girl, so it never worried me. DH is allergic to bee and wasp stings though, so we'll have to keep an eye on the little guy.