What I have found is that when leaned over, your *stomach* muscles are the ones that can best hold you up. Give it a try - when riding, purposely let your arms relax yet keep your upper body leaned over the bars. You will feel your stomach and other torso muscles take up the slack. Once those core muscles get stronger, you should be able to ride farther with much less (if any) discomfort.

(A bike fit is always a good idea too.)