All inline two-wheeled vehicles counter-steer. It's just easier and less dangerous for bicyclists to fight it, than it is for motorcyclists. That has more to do with lower speed than anything else.
If you're interested in the physics, Wikipedia has a decent explanation.
Getting off a bicycle is one of the easier places to observe it, actually. When you're getting ready to stop, turn your bars away from the foot you intend to put down. Your bike will automatically lean toward that side.



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- it happens, especially when standing to climb.
and
