Airplanes turn because they are banked. Pure and simple

Once a bank is established, and all controls neutralized, the airplane will make a nice, 1-gee turn. Now...If you want to maintain altitude, roll in or out of the bank, compensate for engine effects, airframe rigging, and so on, you'll add one or more control deflections into the mix.

An airplane turns when it's banked because the lift vector pulls the airplane laterally, and it weathercocks into the resultant relative wind. Because it all happens together in a dynamic situation, you don't actually see this, but only the result of a smooth turn. You don't need to have any other controls added to make the airplane turn once the bank's established (assuming proper setup, rigging, etc).

Wolfgang Langeweische gives an excellent description of this in his seminal work, "Stick and Rudder".