Glaciers supply sediment to river systems in Alpine regions. Changing glacier dynamics, as a result, impact both the amount of sediment and water entering these river systems. Yet, the hydro-geomorphic processes driving subglacial fluvial sediment transport remain poorly constrained. Fluvial sediment transport in both rivers and underneath glaciers occurs in two primary forms: fine-grained particles carried in suspension and coarser material transported as bedload, intermittently moving along the riverbed. Underneath glaciers, most measured erosion rates evaluate suspended sediment measurements. However, evidence suggests that a significant fraction of sediment leaving the glacier bed is transported as bedload. This study investigates the key mechanisms influencing both transport modes. Using discharge records of bedload and suspended sediment from an Alpine glacier, we calibrate a physics-based numerical model of subglacial sediment dynamics. Our findings reveal that while both transport modes depend on sediment availability, bedload transport is further constrained by subglacial hydraulic conditions, leading to inefficient evacuation. Specifically, the morphology of subglacial channels significantly influences bedload movement. Given the distinct controls on sediment export, glacial erosion assessments must separately account for bedload and suspended sediment transport. Furthermore, understanding how subglacial hydrology and subglacial flow pathways evolve is crucial for predicting climate-driven changes in bedload transport.