Climate controls landscape evolution, but quantitative signatures of climatic drivers have yet to be found in topography on a broad scale. Here we describe how a topographic signature of typhoon rainfall is recorded in the meandering of incising mountain rivers in the western North Pacific. Spatially averaged river sinuosity generated from digital elevation data peaks in the typhoon-dominated subtropics, where extreme rainfall and flood events are common, and decreases toward the equatorial tropics and mid-latitudes, where such extremes are rare. Once climatic trends are removed, the primary control on sinuosity is rock weakness. Our results indicate that the weakness of bedrock channel walls and their weakening by heavy rainfall together modulate rates of meander propagation and sinuosity development in incising rivers.