Simple answer: lift. In typical flight (coordinated) the only force that acts to lift or turn an airplane is lift from the wings, which acts straight up (or perpendicular) from the wings. The tail rudder does not turn an airplane like a rudder on a boat. Instead, to turn an airplane it must raise one wing and lower the other (bank) which alters the vertical component of lift from purely vertical (straight up) to a vector that has vertical and horizontal components (a diagonal). When the pilot pulls back on the stick--increasing lift, the horizontal component of lift changes the direction of the airplanes nose--in effect turning the airplane. An extreme example would be an airplane in 90 degrees of bank or the wings pointing straight up and down relative to the Earth. In this orientation, when the pilot pulls on the stick, increasing lift, all the lift is in the horizontal direction and all the aircraft's lift results in turning the nose of the aircraft horizontally. It's like climbing in the horizontal instead of the vertical.
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