Bridge too high. What should I do?

HI:There is certain amount of adjustment that is normal with a new bridge.  It's difficult to know exactly how high is the bridge without a clinical exam.Best,Dr. Maddahi

It is very common for a new bridge to feel "strange" or "different", especially a long span bridge as you have, but you should not be experiencing pain. 
Without seeing photos of your mouth I can not give you an answer on why the bridge is several mm's higher than the rest of your teeth. There could be several reasons why the bridge was designed this way, with the most common reason being the position of your upper teeth.  
However, a more important question is how the bridge feels when you bite. If you only feel the new bridge when you close then the bite definitely needs to be adjusted or the bridge replaced. A high bite can cause TMJ pain and it will only get worse unless the bridge is adjusted or replaced. 
I would schedule any appointment with the dentist that completed the bridge and/or get a second opinion.