I fell and injured my forearm near the elbow. The X-ray taken at urgent care came negative for fracture. Three weeks later, I can still not fully extend my arm and it pops also. What should I do?

Need new X-ray. Non-displaced fractures may be difficult to see on initial x-rays. I typically will repeat them in 1-2wks. In this case, radial head and/or neck fractures are common. Find a board certified orthopaedic specialist, for further evaluation and treatment.
Find a doctor! If you continue to have symptoms despite a reported -ve xray then please go see a doctor; you may need a repeat film and/or an orthopedist/further work-up!
Need follow up. You can still have a fracture. Radial head fractures can be difficult to see at first. Because you are still having symptoms, follow up with your physician. He/she will like order a follow up x-ray. If that is negative, an MRI may be warranted. Hope you feel better soon.