Originally Posted by AndreyT

Strictly speaking, air drag increases linearly with speed as long as the air flow remains laminar. As the speed increases, the flow eventually detaches and becomes turbulent, and the air drag becomes proportional to the square of speed. In real life it is usually somewhere in between, and for faster speeds it moves closer to v^2.

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A typical bike with a cyclist on it is not a very efficient aerodynamic shape, so it is not a surprise that the relationship will largely depend on v^2 even for low speeds.