The stream power law model has been widely used to represent erosion by rivers but does not take into account the role played by sediment in modulating erosion and deposition rates. Davy and Lague (2009, https://doi.org/10.1029/2008JF001146) provide an approach to address this issue, but it is computationally demanding because the local balance between erosion and deposition depends on sediment flux resulting from net upstream erosion. Here, we propose an efficient (i.e., O(N) and implicit) method to solve their equation. This means that, unlike other methods used to study the complete dynamics of fluvial systems (e.g., including the transition from detachment‐limited to transport‐limited behavior), our method is unconditionally stable even when large time steps are used. We demonstrate its applicability by performing a range of simulations based on a simple setup composed of an uplifting region adjacent to a stable foreland basin. As uplift and erosion progress, the mean elevations of the uplifting relief and the foreland increase, together with the average slope in the foreland. Sediments aggrade in the foreland and prograde to reach the base level where sediments are allowed to leave the system. We show how the topography of the uplifting relief and the stratigraphy of the foreland basin are controlled by the efficiency of river erosion and the efficiency of sediment transport by rivers. We observe the formation of a steady‐state geometry in the uplifting region, and a dynamic steady state (i.e., autocyclic aggradation and incision) in the foreland, with aggradation and incision thicknesses up to tens of meters