Can some planes turn using just the rudder or without banking?
Yes it is possible to turn without ailerons. It is not recommended though.
I have tried it on Cessna 172 as practice many times. I started practicing this maneuver some years ago after I witnessed a disabled Cessna with its ailerons jammed at an angle keeping it in perpetual turn was safely flown back to Van Nuys airport by an experienced instructor on the right seat. when you apply say left rudder the plane will bank left and turn left but you have to control the turn rate by power and rudder. The rate at which one applies the rudder is important; also you should be aware and anticipate potential risk of rolling.
Again it is not recommended but in case of emergency it could save the day!