There are several ways to do so, but generally in bicycling we do turn the bars (else we'd fall).\


Surely you've experienced shifting your center of gravity forward and back, without turning the bars. You shift it up or down by standing up or crouching down. You may shift your center of gravity left by pushing on the right pedal, without turning the bar. Then turn the bar left, and you will have turned left without counter-steering.

There is no cut-off speed for this by the way. Counter-steering "works" primarily by steering the bike over, beneath your body. That's all it is, literally. It's easier to shift the bike because it has less mass (ie, less momentum).