Abstract Understanding spatial drought relationships is important for effective drought relief. The river network—that is, the system of rivers, channels, lakes, and reservoirs within a river basin—influences spatial streamflow drought connectivity and modulates drought characteristics. However, these effects are difficult to disentangle from spatial drought dependencies originating from other hydro‐meteorological processes. Here, we use high‐resolution hydrological model simulations over Europe and study the river network effect in isolation by comparing drought extents of unrouted runoff to those of routed discharge. We find that the river network increases drought extent in most European basins. Runoff droughts in snowmelt‐dominated upstream regions often lead to larger discharge droughts than the original runoff drought, whereas discharge droughts in downstream rivers are affected by runoff droughts upstream. Water bodies influence the river network effect by storing water, thereby decoupling runoff from discharge. Our results can help water managers to better understand spatiotemporal drought development.