The dynamics of a turn involve bank angle AND the weathervaning effect of the vertical fin (the rudder has a secondary function) (plus pitch, but we'll leave that part out to keep it simple).
When you bank, the wing's lift vector is tilted, and the lateral force introduced by the tilt drives the aircraft sideways (it sideslips). Without a weathervaning effect, it would move laterally while remaining pointed in its original direction; just continue sideslipping. This sideslip happens initially (briefly) when bank is first applied, and is critical for dihedral effect to work for roll stability.
The result of the sideslip is a lateral angle of attack acting on the airplane's vertical profile, resulting from the sideways movement created by the bank, mostly acting on the fin, or more correctly acting on the vertical aerodynamic center, the vertical Neutral Point, of the entire aircraft. As long as the Neutral Point is aft of the airplane's C of G (what the fin is there to do), there is a positive weathervaning tendency, or positive static stability in yaw.
So the bank makes the plane slew sideways, the fin develops lateral lift in response (after a slight lag to allow dihedral effect to do its thing in the event you didn't actually want to bank), and rotates the body about the yaw axis to reduce the angle of attack on the fin to near zero.
So the airplane banks, starts to slew sideways, but immediately afterward the weathervaning effect of the fin rotates the body in yaw to keep the body aligned into the airstream and the sideways motion is accompanied by a rotation that makes the airplane follow the arc of the turn.
What the rudder does is allow the camber of the vertical fin to be varied by the pilot (being more or less a wing flap that works in both directions), to apply forces beyond the basic weathervaning tendency, when the weathervaning tendency is insufficient. What makes the weathervaning tendency insufficient is aileron adverse yaw.
So rudder application is used to adjust the camber of the vertical tail, to apply force beyond the basic weathervaning effect, to cancel out the yawing moment created by adverse yaw from the ailerons. If the ailerons are neutral while banked, little to no rudder is required because there is no adverse yaw.
So for a turn:
- Bank creates a lateral thrust component to move the plane sideways.
- The vertical fin responds to the sideways movement by continuously weathervaning the body into the lateral airflow created by the bank. The fin needs to be small enough to allow a slight lag in this reaction to allow dihedral effect to work, where the bank was induced by a bump and the objective is to make the plane return to level flight on its own, but large enough to give a positive weathervaning tendency beyond that when the bank is deliberate.
- The rudder allows the camber of the vertical tail to be varied to increase or decrease the weathervaning force, mostly to cancel out the yaw forces induced by the aileron displacement. The rudder can also be used to take out the small lag in weathervaning effect you normally get when you bank as mentioned previously. Rudder application is normally roughly in proportion to the up aileron displacement. Airplanes with rudder interconnect systems, like the Piper Tripacer, mechanically directly gear rudder movement to aileron movement (using bungee springs) and the pilot doesn't need to move the pedals to maintain a coordinated turn - left aileron gives left rudder and right aileron gives right rudder and neutral aileron gives neutral rudder.