Is there a way to get rid of flea bites quickly?

Flea bites. Use antiseptic soap to wash the flea bites. Do not scratch flea bites. You can use 1% Hydrocortisone cream on the bites or take antihistamines for itching. A paste made of 2 parts baking soda ; 1 part water can be applied to bites. The bigger issue is that you must get rid of the fleas. Many people elect to hire an exterminator.
Sure! Treating the source of the fleas (cat or dog) is the most important step. Fleas rarely live on humans. They take blood meals and leave. The itchy bumps are your immune system responding. You can try an over the counter oral antihistamine for itch and a topical Hydrocortisone to apply to each lesion for a few days. However, remember this just helps your symptoms and does not kill the fleas.