This is largely a myth. A rotating bicycle wheel does generate a small amount of gyroscopic effect, but it is too small in comparison to the weight of bike & rider to make much difference. People have built bicycles with counter-rotating front wheels[1] to prove it.

Smaller bikes may feel less stable, however I suspect this is simply because smaller wheels will be affected by smaller road imperfections that a larger wheel would simply roll over.

[1] https://theconversation.com/how-does-a-bike-stay-upright-sur...