Large alluvial rivers transport globally significant quantities of water, sediment, and nutrients to the oceans, temporarily storing and cycling this material within the bars, islands, and floodplains that define their morphology. The world’s largest rivers display a remarkable variety of morphologies. However, existing theory and numerical modeling fail to explain this diversity, which remains poorly understood. This study applies a new numerical model of water flow and sediment transport to show how the morphology of large sand-bed rivers is influenced by bed sediment mobility, bank erodibility, and rate of floodplain development. Simulations demonstrate that a wide range of river styles, including meandering, anabranching, and braiding, can occur over a relatively narrow range of environmental conditions. Results highlight the suspension of bed material, which limits the gravitational deflection of sediment in the direction of the local bed slope, as a key control on sediment transport direction and hence river morphology. Moreover, high mobility of bed and bank sediments are hypothesized to favor contrasting river styles, although both may be promoted by increasing stream power. These results explain the inability of existing stream power theory to predict the morphology of the world’s largest rivers, and highlight the potential for investigating river-floodplain co-evolution using physics-based simulation models.