[nathanielc]

Bike are balanced and symmetric side-to-side. Top to bottom they are not, obviously. The center of mass is above the ground , so a bike that is not moving will fall over to lower its center of mass. But this doesn't happen if the bike is moving. Push a bike without a rider and it will travel for a few feet without falling over, and take longer to fall over if just left standing still.

[Someone]

The surprising thing is that it is way more than "a few feet". Apologies for repeating myself, but watch https://www.youtube.com/watch?v=oZAc5t2lkvo (And that is not on smooth terrain)

And nitpick: typical bicycles aren't symmetric left-right. The chain and gears make them heavier at the right. Most mass is close to the center line, though, so its moment is negligible, especially when ridden by a human.