Why do steep rivers erode their beds more quickly than gentle rivers?

Steep rivers erode their beds more quickly than gentle rivers primarily because of the relationship between slope, flow velocity, and energy.

In geomorphology, a river's power to erode and transport material is derived from the conversion of potential energy into kinetic energy as water moves downstream. The velocity of a river’s flow is largely determined by the slope of its channel; a steeper slope results in higher flow velocity. This increased velocity enhances the river's capacity to transport sediment, which acts as the "tools" used to carve and incise the riverbed.123

The power per unit length of a stream ($\Omega$) can be expressed as:
$$\Omega =\rho gQS$$
where $\rho$ is fluid density, $g$ is the acceleration of gravity, $Q$ is flow discharge, and $S$ is the slope. Because the stream gradient ($S$) is a direct factor in this equation, a steeper slope increases the available power for incision at a given discharge. Consequently, rivers with steeper gradients possess greater energy to overcome the resistance of the bedrock or channel materials, leading to more rapid erosion compared to rivers with gentler slopes.23