Is it normal to have a giant, swollen, red, warm, bump on my arm after allergy shot?

Not unusual . This can happen. Discuss this with your physician. Local reactions can occur. If bothersome, modifying doses of the injection can help. Sometime taking an antihistamine 2 hours before injections can help. If you have any trouble breathing, feeling lightheaded, you must consult you physician or go to an emergency room as can be a more serious reaction.