I seem to sweat much more than most people when I exercise. Is there a medical explanation for this?

Excessive sweating. If you're sweating more than normal with exercise, you may be "out of shape" and when you've been exercising the body is trying to regulate the core temperature of your body. If this is not the situation for you then maybe your thyroid is out of balance. It may, however, be normal for you if this has been since you've young.
Hyperhidrosis Sweat. Hyperhidrosis (or excess sweating) is more common than you'd think! antiperspirants are the simple answer, but often won't work with true hyperhidrosis. I have had success treating patients underarms, hands, feet, forehead, and scalp with botox. Botox is fda approved for sweating and is very safe and effective. Botox typically lasts for 6 months when used for excess sweating. Hope this helps!