Abstract Given the insurmountable challenge of measuring all rivers in situ, global river models serve as the foundation of freshwater knowledge past, present, and future. We adopt an inductive empirical framework based on Surface Water and Ocean Topography (SWOT) satellite measurements to assess these models. After controlling for SWOT data quality, we examine 68,347 individual river reaches representing ∼38% of global discharge. We find river models currently struggle in areas of heavy economic development, multi‐channel rivers, arid areas, and many Arctic rivers. After controlling for these expected errors, we find better skill as rivers get wider and that parts of Siberia and China are particularly difficult to model. We also find large variability and spatial heterogeneity to model performance, resisting oversimplification. Our results suggest that leveraging SWOT observations within river models will improve them, but river models must adapt their structure to represent realistic hydraulics to do so.