Spatial and temporal variations in rainfall are hypothesized to influence landscape evolution through erosion and sediment transport by rivers. However, determining the relation between rainfall and river dynamics requires a greater understanding of the feedbacks between flooding and a river's capacity to transport sediment. We analyzed channel geometry and stream-flow records from 186 coarse-grained rivers across the United States. We found that channels adjust their shape so that floods slightly exceed the critical shear velocity needed to transport bed sediment, independently of climatic, tectonic, and bedrock controls. The distribution of fluid shear velocity associated with floods is universal, indicating that self-organization of near-critical channels filters the climate signal evident in discharge. This effect blunts the impact of extreme rainfall events on landscape evolution.