What insect could cause my daughters forearm to swell 4 inches around the bite?

Several. It may not be specific to the insect. The body often responds to a variety of insect bites by trying to build a wall of fluid around any injected material. Capillaries under the skin turn into soaker hoses and release serum & chemicals that cause swelling. A warm thichened area builds similar to what happends when u sprain an ankle.The swelling goes down as the lymph glands sort out & detoxify it.