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.

The most widely used is 'I turned the engine off', they are all correct but this is most common and you will sound the most native if you say that.