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Let's say we have a bike pointing forward in the $+z$ direction into the screen. It's moving in the $+z$ direction hence the wheels have angular momentum in the -x direction ,which is towards the left of your screen. Now when we turn the wheel right we apply a torque in the -y direction. As the change in angular momentum must be in the same direction as the torque why doesn't the wheel's angular momentum "follow" the direction of torque. Kind of like in the following video https://youtu.be/XPUuF_dECVI at time 19:17.

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The bicycle wheel is not in a gimbal mounting, the bicycle wheel is constrained. For that reason alone we should not expect the front wheel in the front fork to move the same as a gimbal mounted gyro wheel.

For more information about bicycles: video on the Minutephysis youtube channel: How do bikes stay up?

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