The wave attenuation properties of seagrasses are key to accurately predict how effective these plants are at protecting coasts from erosion and floods. While recent studies have significantly advanced the understanding of seagrass wave attenuation in pure-wave conditions, the presence of a current introduces several complications that have yet to be fully explored. In the present study, we quantify the wave attenuation of seagrass canopies in the presence of a current parallel to the direction of wave propagation via experiments conducted with dynamically scaled mimics of seagrass installed in a laboratory flume facility. The dataset we present is the largest of its kind and spans a broad range of wave properties, current velocities, water depths, and plant densities for a total of over 300 experiments. Using our experimental results, we show that the commonly employed approach of modeling wave attenuation as a result of vegetation drag works well for a range of conditions but underpredicts systematically when turbulence generated by the interaction between the seagrass canopy and the current is sufficiently strong. We then employ phenomenological arguments and experimental data to identify a nondimensional parameter that effectively quantifies the relative importance of turbulence and drag in dictating the overall observed wave attenuation. Moreover, we propose a simple but physically based modeling approach that is consistent with the proposed phenomenology and can be used for applications in coastal waters.