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This question popped up on Quora this morning and it made me think, so I've copied my response for your amusement. Some of the other answers mentioned counter-steer, so I included my take on that subject as well.
Q: Why does a motorcycle turn when leaned over?
A: Good question. It doesn’t actually. You can lean a bike over and keep going in a straight line, but you need to shift your weight in the opposite direction to keep it balanced. But you are asking about turning, so this doesn’t matter.
I suspect that leaning and turning go together because of the profile of the tire. The outer edges of the tires are smaller in diameter than the center. When you are riding straight up the tires act like a cylinder and so roll straight ahead, like a can rolling across a table. When you lean the bike over, the tire becomes more like a cone and so it starts rolling in a circle, kind of like a funnel rolling across a table top.
I road for several years before I discovered the trick of opposite steer. Not sure how I managed to get around corners before that. In any case, you don’t actually move the bars, all you need to do to turn right is to push on the right hand handlebar. If you are traveling at any speed, like 30 MPH or better, the bar won’t actually move, but the effect of pushing on the handlebar is instantaneous and you will go zooming off to the right.
What is happening is the gyroscopic action of the front wheel is reacting to the pressure of your hand and pushing the bike over, which makes the wheels turn into cones which makes your bike follow the curve of the road.