In this paper we consider wireless mesh networks (WMNs) used to share the Internet connectivity of sparsely deployed fixed lines with heterogeneous capacity, ranging from ISP-owned high-speed links to subscriber-owned low-speed connections. If traffic is routed in the mesh without considering the load distribution and the bandwidth of Internet connections, some gateways may rapidly get overloaded because they are selected by too many mesh nodes. This may cause a significant reduction of the overall network capacity. To address this issue, in this paper we first develop a queuing network model that predicts the residual capacity of network paths, and identifies network bottlenecks. By taking advantage of this model, we design a novel Load-Aware Route Selection algorithm, named LARS, which improves the network capacity by allocating network paths to upstream Internet flows so as to ensure a more balanced utilization of wireless network resources and gateways’ Internet connections. Using simulations and a prototype implementation, we show that the LARS scheme significantly outperforms the shortest-path first routing protocol using a contention-aware routing metric, providing up to 240% throughput improvement in some network scenarios.