The distribution of quantum entanglement between remote clients offers opportunities for secure communication, distributed quantum computation, and quantum-enhanced sensing. A key technology for allowing such a quantum network to extend over large distances is the quantum repeater, which typically employs a long-lived quantum memory with an optical interface. The past decade has seen tremendous progress in experimentally realizing the fundamental building blocks of these repeaters, but open questions remain on what architectures achieve optimal entanglement flows in multi-user quantum networks. Here, we propose a quantum router architecture comprising many quantum repeater nodes in a flexible photonic switchboard architecture to broker entanglement flows across such networks. We compute the rate and fidelity of entanglement distribution using this architecture using an event-based simulation platform. We find that the router's photonic switchboard improves the entanglement fidelity with the number of qubits per node, without a significant drop in the entanglement distribution rate. The proposed quantum router architecture uses present-day photonic switch technology such as photonic integrated circuits, opening a path to near-term deployable multi-user quantum networks.