I am relatively new to mopeds but I've been riding motorcycles, big and small, for over four decades. For nearly 20 years I've been a nationally certified motorcycle skills instructor. I am going to go out on a limb and say mopeds aren't much different than motorcycles. The physics are the same, even if the speeds and power output are not.
When I read things like "I had to lay it down" or "give up and head for the curb", I cringe. The object of riding is not crashing. Motorbikes of all sizes stop and turn a lot faster on their tires than on their sides. If you're coming into a curve at a speed you sense to be "too fast", don't give up. You may be out of your comfort zone but you're probably not out of traction. You almost always have more traction and ground clearance than you think. Look into the curve and press harder on the inside (low) bar. Press right, go right. Press left, go left. Don't ever give up, ride it and you'll probably be OK. Now I will put away the soapbox. Thanks.