I don't think anyone has hit on what is really going on here. The
explanations about the sideways force created by lift when the AC is banked
cause it to move ( accelerate ) to the side, but not rotate. The dihedral
wing explanation doesn't work either, because the raised wing's horizontal
force is applied aft of the CG, thus causing rotation in the opposite
direction of the turn. The major reason that a plane rotates about the
vertical axis during a turn is wind vaning.

You bank the plane, the lift is broken into horizontal and vertical
components. The horizontal force causes the plane to accelerate to the
side, but not rotate as others have stated. Now with the plane picking up
speed in the lateral direction, the relative wind is now coming from one
side of the plane, a forward quartering headwind! As anyone who has taxied
on a windy day knows, planes have a natural tendency to face into the wind.
This is caused by the big wind vane we call a tail. This rotates the plane
into the wind. The relative wind is thus always leading the plane by a few
degrees, causing a continued rotation. And as a side benefit, the
horizontal acceleration is countered by the centripetal force of the turn,
so we don't continue to accelerate to faster and faster horizontal speeds.