Is the turning motion of a banked airplane caused by true centripetal force?
Centripetal force is defined as the component of the total force acting on the object that is causing it to follow a circular path. It is the force exactly perpendicular to the velocity vector.
To create a circular path, the force needs to turn with the velocity. If you have a force that keeps its spatial orientation, less of it will be centripetal and more of it will just accelerate the object as it moves and the resulting path will be parabolic (e.g. the ballistic curve under gravity).
The horizontal component of lift always contributes wholly to the centripetal force, but this is again quirk of the definitions: lift is the component of the aerodynamic force on the wing that is perpendicular to the relative wind and drag is the component that is parallel to it. So analysing the situation in the reference frame of the air mass, relative wind is the velocity vector.
Now if the plane didn't turn with its velocity vector, it would just gain side-slip, the lift would decrease—banking into side-slip reduces the angle of attack—and the aircraft would stop turning—it's flight path does turn initially, even if the body does not—and settle in sideways flight. So we need to make the aircraft rotate with the turn.
Now just like linear motion, rotating motion stays constant¹ unless acted upon by a moment of force (a.k.a torque). So when initiating the turn, a yawing moment of force needs to be created to also initiate the rotation.
A moment of force may be created by a pair of force of opposite direction and different action lines, so it does not have to cause any linear acceleration, but in case of aircraft usually an initially unbalanced force is created on the tail that starts the rotation, but also accelerates the aircraft sideways until balanced by drag on the fuselage. Note that this force is out of the turn (and up), so it reduces the centripetal force a bit.
But in a steady turn the torque is zero. Or oscillating around zero as the correct angular velocity is maintained via feedback loop. So the horizontal component of lift is the only centripetal force in a coordinated turn. In not coordinated turn, some side force on the fuselage is added or subtracted depending on whether it is a slip or skid.

¹ Angular momentum is conserved but the angular velocity does not have to be. Depending on the axis of rotation and mass distribution (moment of inertia) you may get strange phenomena like the Dzhanibekov effect.