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I believe that the answer to this lies in the forces acting on the bike, namely the the frictional force, the turn itself is made by the wheels, which create friction sideways. The centripetal force acting in the turn is the friction, imagine a motorcycle driver that doesn't lean into the curve; because he doesn't lean, the friction is not strong enough to keep the driver from going in a straight line. However, when he leans, the static friction from the road increases, therefore the centripetal force increases.
Note: There is an aspect of this question that also relates to Torque.
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