Let me do a bit of correcting....
So gyros are in effect, but mainly to slow your steering more than to cause it to happen.
You'll note that you can countersteer fast at very low speed (even though gyros are very weak). The key is that turning the wheel in one direction sends the front of the bike to the side, but momentum is still going straight forward. This means that the direction of the bike and the center of gravity are offset and the bike tries to fall on its side. Luckily, once you stop counter-steering the offset of the front wheel causes the front tire to turn into the corner, which then settles the bike again and everything stabilizes.
As for turning, you get some of it from the camber of the rear wheel when leaned over, but the front helps point the bike some as well (you can tell this mostly if you try to countersteer a little bit while mid-corner -- the bike resists and tries to run wide a bit). You can also see this if you push the front too hard into a corner, the front will push and you will run wide. If all your turning force came from the rear, the front tire would be relatively irrelevant and losing the front wouldn't happen.