The bike analogy is alright for the purpose of showing that a turn should be initiated with the front foot, because a bicycle cannot turn with its front wheel off of the ground.
However, as others have said, bicycles use countersteering to initiate turns. The problem with trying to tell people about this is that it isn't something you often do consciously on a bike, you just do it. When I was getting my motorcycle license, however, I really became aware of countersteering. The basic premise is that in order to turn a motorcycle, you need to lean in the direction of the turn. The problem is you can't just start leaning to one side or you're going to crash; the bike can't stay upright. To initiate a right turn on a motorcycle by leaning, you first push on the right
hand grip to steer the front wheel to the left. At this point, you begin to lean to the right, and then finally apply pressure to the left hand grip to keep the front wheel pointed in the desired direction (to the right because you're turning right).
A decent explanatory video, although countersteering can occur at speeds well below 15 mi/hr, contrary to what the gentleman says: YouTube - Swerve Technique on Motorcycle (Countersteering)
This one is sort of a lame explanation, as anyone who has taken a basic physics class knows that "centrifugal force" is a fictional force. Watch the low-speed demonstration though: YouTube - Slow-speed countersteering
Countersteering occurs when you are on a bicycle also, but it's less noticeable because the bicycle has much less inertia than the motorcycle. The bicycle is much easier to lean than the motorcycle, but a slight press on the right handlebar is still the best way to initiate a right turn.