> turns that I make do not involve turning my front wheel at all (in the sense that: I don't move my hands consciously).

In all likelihood you are doing it unconsciously, generally, as one goes to initiate, say a right-hand turn, as you begin to lean right, you also apply a slight additional pressure to the right handle bar, which steers the wheel to the left, this initiates counter-steer which leans the bike to the right. With a stable bike, and a fixed front fork, it would be exceedingly difficult to initiate a lean significant enough to navigate most roads at the speed that we do.