At some point, someone noticed racing bikes counter steer really severely so you can easily see the wheel is actually pointing the opposite way. But the reality is that even at "won't fall over" speed on a bicycle, you're already doing the exact same thing. When you lean, you counter steer. Otherwise you'll high-side like a missile.
> I have asked dozens of bicycle riders how they turn to the left. I have never found a single person who stated all the facts correctly when first asked. They almost invariably said that to turn to the left, they turned the handlebar to the left and as a result made a turn to the left. [...] I have never found a non-scientific rider who had particularly noticed it and spoke of it from his own conscious observation and initiative.
The existence of counter-steering is still controversial to some riders, to the point where machines like the "No B.S. Bike" were created to demonstrate it as a necessary effect: https://soundrider.com/archive/safety-skills/nobsbike.aspx
To turn, you lean in the direction you want to go. But what way does the handle bar turn when you do that? It counter steers! You will fall if it doesn't (which is essentially what the No B.S. Bike demonstrates).