The counter steering answer to that question is rake, the fact that the steering axis hits the ground in front of the contact patch if the front tire, so that when a bike is leaned over it self-steers into the turn (the contact patch torques the wheel around the steering axis) and countersteers itself upright.

It's well known in motorbikes that the amount of rake is what makes a bike more stable vs more responsive to steering. Some folks lower the front of the bike - tighten the triple tree lower on the forks - to increase responsiveness on otherwise more boring bikes.

Of course there's more to it all, gyroscopic effects and so on, but this is the countersteering perspective.