I should preface that this is from an American English perspective (New York, northeast US). Your mileage may vary.
I shut the engine - not common
I switched off the engine - okay
I shut off the engine - okay
I switched *off* the engine - okay
I turned off the engine - I would use this (especially for a car ignition)
"Switch off" has the image of a toggle action, like in a light switch, so it wouldn't sound odd, but there is a nuance in the movement/motion of the action.