How could I heal bursitis and tendonitis in my elbow?

Stretching. Stretching exercises by keeping the elbow fully extended and then bending the wrist up and down are the best way to stretch the tendons of the elbow. Alternating heat and ice to the affected area can also help. If this does not help an injection sometimes is used in the most severe cases even surgery may be needed.
Capsaicin patches. You can also use local capsaicin patch upto 16 hours and wrap with ace bandages for comfort.

Related Questions

What to do if I am wondering how you can tell the difference between a bursitis at the elbow and tendonitis at the elbow?

The location. The bursa or fluid filled sac is located over the olecronon process. This is the tip of the elbow which many people lean on. Because of trauma it often gets inflamed. Tendinitis typically occurs on either side of the elbow, close to where the wrist flexor tendons (medial epicondylitis) or wrist extensor tendons (lateral epicondylitis) attach to the bone.