Magnetic reconnection is a universal process that powers explosive energy-release events such as solar flares, geomagnetic substorms and some astrophysical jets. A characteristic feature of magnetic reconnection is the production of fast reconnection outflow jets near the plasma Alfven speeds(1,2). In eruptive solar flares, dark finger-shaped plasma downflows moving toward the flare arcade have been commonly regarded as the principal observational evidence for such reconnection-driven outflows(3,4). However, they often show a speed much slower than that expected in reconnection theories(5,6), challenging the reconnection-driven energy-release scenario in standard flare models. Here we present a three-dimensional magnetohydrodynamics model of solar flares. By comparing the model predictions with the observed plasma downflow features, we conclude that these dark downflows are self-organized structures formed in a turbulent interface region below the flare termination shock where the outflows meet the flare arcade, a phenomenon analogous to the formation of similar structures in supernova remnants. This interface region hosts a myriad of turbulent flows, electron currents and shocks, crucial for flare energy release and particle acceleration.