Is it possible to have bursitis and kill the pain?

Unsure of question: Yes, the pain from bursitis can be alleviated. Typically it's treated with rest, ice and NSAID meds sometimes when especially painful a cortiosone injection can be very helpful. Also while slower in response physical therapy can help.

Yes: You need a correct diagnosis, but after that specific therapy to that diagnosis can help. It may be exercise and physical therapy, it may be heat, ice or ultrasound; it may be a localized injection. These are some of the approaches.