We report experimental measurements of the onset of grain motion in fully submerged erodible beds containing grains of two sizes driven by a turbulent shear flow. Although we find that a traditional Shields-number framework successfully accounts for grain-size effects in beds composed of grains of only a single size, our results show that bimodal beds are distinct in a way that cannot be captured by a Shields-number rescaling. In particular, we find that large grains in a bimodal bed are mobilized into bedload transport by stresses that would normally be subcritical for them but that the behavior of small grains is less significantly affected. By analyzing higher-order statistics, we reveal the key role played by the granular contact and force networks in the bed near onset and clarify the impact of grain-size polydispersity on the transition from a static to an eroding bed. Our results have implications for modeling sediment transport in natural systems where polydispersity is unavoidable.