As well as bicycle racing, I did some motorcycle racing in my younger days. "Steering" a motorcycle is counter-intuitive. One of the firs things I was taught on the track was how to do a "flick", which is a quick turn in one direction or the other. To make a quick right turn, you actually push against the right grip and push the foot peg with your left foot, this causes the motorcycle to lean quickly and turn to the right. To make a quick turn to the left, you do the opposite. Once you are leaned over in the turning position, you more or less naturally maintain the balance necessary to navigate the turn, faster or sharper turns requiring a steeper lean than slower or more gradual turns.
Counter-steering is something one does when the rear wheels and the front wheels are attempting to go in different directions, such as when you lose control of a car (or motorcycle) on a slippery service. The back wheels slide left and/or right, and you turn the wheel or bar in order to make the front wheel/s counteract the direction of the rear wheel/s. In such situations, you control the overall direction by turning the front wheel/s in the direction you want to go, and controlling the direction of the rear wheel/s with the throttle. You'll see this in flat track motorcycle racing, or race cars "drifting." More throttle pushes the rear of the vehicle more outward, less throttle moves the rear of the vehicle inward. With no throttle, the rear wheel/s will follow the front wheel/s. On a bicycle this is irrelevant, as your legs cannot provide power as smoothly or as strongly as an engine.