Nowt wrong with a 500D and 70-200L...
Silverstone is a pain unless you have a long lens, with huge run offs it makes it difficult to get close, try cropping first, should have some leeway with 15mp. Try going to smaller circuits to hone your skills (Oulton Park, Cadwell etc) as you can get closer and the lack of a long lens is more easily negated.
Some techniques to try:
1. Prefocus on a point on the track at f8 - drivers try and use the same bit of track so this would give you some margin of error with depth of field, remove focusing inconsistencies and allow you to visualise the shot.
2. When panning a longer shutter speed gives more impression of speed - keep panning after the shutter has finished, this will help with smoothness.
3. If you can get close to the fence with the 70-200 use it wide open and prefocus on the track, the fence will pretty much disappear due to depth of field, it's a bit old at first but can get you some great shots.
4. Pick a spot on the track, with good access, with an elevation change, this can help with drama, another option is heavy braking zones that often cause lock ups ( huge puffs of smoke look good
)
I agree with the previous critique, too many angles, a bit distant and they don't look super sharp. Hope this helps, keep trying mate!