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.

For every car, for example, there's a more-or-less well defined threshold speed, over which the turbulent component starts to dominate (the exact value depends on the aerodynamic properties of a specific car). Under that threshold the car's fuel consumption (MPG) depends very little on the actual speed. Over that threshold, as v^2 component begins to become more prominent, the car's MPG is dropping noticeably as the speed increases. It is often assumed that the threshold value is 55 mph, while in reality the number makes little sense, since for each car model it is different (sometimes significantly).

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. A trailer can be built in a much more efficient aerodynamic shape, thus extending the range of speeds in which the linear air resistance component dominates over the squared component. This is splitting hairs, of course, especially if the trailer is pulled by a bike (which immediately negates any aerodynamic benefits).

In any case, to call the relationship "exponential" (as previous poster did) is certainly a major error.