How is stream discharge related to channel size/shape?

During time of high discharge the river becomes wider, deeper and rounder. As the discharge has increased the water will flow faster and have more energy. The increase in energy means there is more friction with the bed and banks of the river, causing them to erode, making the river wider and deeper. The increase in energy also means that the water can carry more, and larger sediment which further erodes the bed and banks the river.
The opposite is true when the discharge is low, as the water will have less energy and therefore, less erosion of the river bed and banks will take place, making it shallower and narrower.