How to remove governor from Suzuki LTZ-50
Yes, remove the secondary pilot jet. Make sure both intake and exhaust restrictors are removed. Remove the end cap from the muffler exhaust pipe (held on with two screws). Drill a few extra holes in the airbox lid (the existing hole is smaller than the carb throat, thus restricting airflow). Finally, replace the main jet with a #75 Mikuni jet, available from any Suzuki dealer for a couple of bucks. Make sure to readjust the idle back to 1800 RPM when you're all done.
With these changes the little engine will wake up quite a bit. My son can sustain ~20 MPH on normal trails and has enough torque to climb almost anything. He absolutely loves it.
You can go farther, using aftermarket exhaust systems and shaving the head and smoothing the ports and retrofitting larger carbs and such, but those take more time and cash. The changes recommended above won't cost $10 total. Think of it as the low hanging fruit waiting to be harvested.