[dahart]

Try doing it consciously! It's very fun & informative. Try not to lean, and do some experimenting while you're riding fast. The faster you go, the more noticeable the effect is. You can push forward on the right side, and it will immediately put you and your bike into a right lean. Hold your hand pushing slightly on the right side to continue turning right. The handlebars will push back a little and try to straighten out.

Many people that ride bikes never know about counter-steering, unless they ride motorcycles too. So, you're not alone. I didn't know about it on a bike until I learned riding motorcycles.