Here is an excellent demonstration of how strong the force can be by the folks at MIT:
A simple experiment you can do at home with very little prep is to balance a penny on its side on a table and bang on the table. The penny will fall over. Now, roll the penny across the table and bang on the table while it's rolling, It will roll merrily on its way. The angular momentum of the rolling penny allows it to easily overcome the vibrations in the table.
Basically, the faster the wheels are spinning, the greater the tendency of the bicycle to stay upright and the less you have to shift your weight and make corrections with the steering to maintain your balance. As the wheels slow down, the force is reduced and your dependence on on true balance and steering increase.