Let's suppose that you were able nearly instantaneosly make a ninety degree turn with the bike steering. Imagine what would really happen. The bike's front wheel will not be able to continue to roll because it will be perpendicular to the motion of the bike. If the coefficient of static friction is large so that it is not exceeded then you will be flung out of your bike seat and the bike will flip over. I think i've actually had this happen to me in real life when I was younger. But the front wheel probably was sliding a bit (meaning that the static frictional force was overtaken and kinetic frictional force began to be excerted on the tire from the ground.) And I rotated forward with the bike before I flew off. It is similar to what would happen if you applied brakes only on the front wheel.
You will have to take into account the frictional force of the road on the "turned" wheel.
By the way- a really nice thing happens with bicycles! If you tilt the bicycle to the right, the torque on the front wheel will automatically turn the wheel in that direction and then entire bicycle follows, preventing the bicylcle from falling over. That's why you can ride a bicycle (or even more obviously a unicycle) "no hands". It's not necessary to actually "turn" the handlebar- that will happen automatically. You really steer a bicycle or motorcyle (and, again, even more obviously a unicycle) with your body, not your hands.
If you are concerned about conservation laws, look at it this way. The turning process is relatively lossless, so the vehicle's initial and final speeds must be the same. The momentum (vector) is not the same before and after, because the momentum is trnasferred to the earth.
If you want to understand by looking at the mechanical details, look at it this way. The wheels can only transmit a force parallel to their axes, since they are free to spin in the perpendicular direction. So when the front wheel is turned, there is a force perpendicular to the vehicle's direction. This turns the vehicle in exactly the same way that a string tied to a mass and fixed at one end causes the mass to travel in a circle.
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