Bursae that are not infected can be treated symptomatically with rest, ice, elevation, physiotherapy , anti-inflammatory drugs and pain medication. Since bursitis is caused by increased friction from the adjacent structures, a compression bandage is not suggested because compression would create more friction around the joint. Chronic bursitis can be amenable to bursectomy and aspiration.  Bursae that are infected require further investigation and antibiotic therapy. Steroid therapy may also be considered.  In cases when all conservative treatment fails, surgical therapy may be necessary. In a bursectomy the bursa is cut out either endoscopically or with open surgery. The bursa grows back in place after a couple of weeks but without any inflammatory component.
I have some info. I developed this same ileopsoas tendinitis two weeks after left hip and two weeks after right hip replacement. The surgeon denied it could be the hip causing this. He sent me to a rheumatologist and low back surgeon, wasting my money. He is considered the best surgeon in the state! He got so frustrated with me. The pain became so severe I could barely walk. I was in misery. Finally, he put cortisone in the trochanter bursa and the pain went away in a couple days. When the same pain occurred after the second hip, I got the cortisone. I have also tried cortisone directly into, by ultrasound, the tendon. That did not work any better than just doing the bursa. So every three months for five times each hip, I got cortisone. I went to PT and stretching made the pain severe and debilitating. I also had several dry needling from the PT which did nothing.
NSAIDs, either selective cyclooxygenase 2 inhibitors (coxibs) or nonselective agents, are effective in treating episodes of nonseptic bursitis. It is prudent to avoid use of NSAIDs and coxibs in patients with known coronary artery or cerebrovascular disease, or those with multiple risk factors for cardiovascular disease, those with renal disease, and edematous states due to the increased risk of adverse cardiovascular side effects and/or the risk of acute renal failure. NSAIDS should also be avoided in patients with history of gastric ulcers and gastro intestinal bleeding.