The shape of most animal cells is controlled by the actin cortex, a thin network of dynamic actin filaments (F-actin) situated just beneath the plasma membrane. The cortex is held far from equilibrium by both active stresses and polymer turnover: Molecular motors drive deformations required for cell morphogenesis, while actin-filament disassembly dynamics relax stress and facilitate cortical remodeling. While many aspects of actin-cortex mechanics are well characterized, a mechanistic understanding of how nonequilibrium actin turnover contributes to stress relaxation is still lacking. To address this, we developed a reconstituted in vitro system of entangled F-actin, wherein the steady-state length and turnover rate of F-actin are controlled by the actin regulatory proteins cofilin, profilin, and formin, which sever, recycle, and assemble filaments, respectively. Cofilin-mediated severing accelerates the turnover and spatial reorganization of F-actin, without significant changes to filament length. We demonstrate that cofilin-mediated severing is a single-timescale mode of stress relaxation that tunes the low-frequency viscosity over two orders of magnitude. These findings serve as the foundation for understanding the mechanics of more physiological F-actin networks with turnover and inform an updated microscopic model of single-filament turnover. They also demonstrate that polymer activity, in the form of ATP hydrolysis on F-actin coupled to nucleotide-dependent cofilin binding, is sufficient to generate a form of active matter wherein asymmetric filament disassembly preserves filament number despite sustained severing.