The bike turns because the front tire points in a slightly differnt direction than the rear. There's some camber thrust effect, but not much. Motorcycle tires don't all have circular cross sections, some older ones (Dunlop K81's) had high narrow domes with nearly flat and very steep sides. Modern Dunlop tires generally have near flat centers (for longer wear), and rounded sides. Bridgestones are pretty close to a circrular arc. You feel the difference. The Dunlops flatter profile resist the initial lean on a motorcycle a bit, transition from a Dunlop to a Bridgestone and it feels like the back end is slipping out from underneath you (no lean resistance, so it takes less countersteering). Racing tires are generally like the Bridgestones, close to a circular arc.As far as I know the bike turns because of camber thrust.
The tires also have a bit of a overhanging lip at the edge of the tread above the sidewall, that allows the contact patch to flex a bit on the edge for better grip (only racers probably notice this).