No problem leaning a bike in motion without turning the front wheel, all you have to do is shift your balance. It won't turn though, just go straight while leaning left or right.
Having said that, it's 100% the case that a 2 wheeled bike with rounded tires in a lean will turn. Just consider that the circumference of the tire shortens as you approach the edge, it must travel a shorter distance per revolution. so without sliding, it clearly needs to turn.
In-fact, at speed, the lean of a bike is the primary factor that contributes to the turn.