Wireless mesh networks have been attracting many users in recent years to achieve a wide-area wireless environment with flexible-configuration and low-cost properties by connecting base stations (mesh nodes) with each other via wireless connections. When the wireless mesh networks are used as an infrastructure for Internet access, all network traffic from mobile nodes go through a gateway node which is directly connected to the wired network. Therefore, we need to distribute the entire traffic load by deploying multiple gateway nodes and each mesh node should select a gateway node to access the Internet according to its traffic load, processing power, and access link capacity. In this paper, we propose a routing method that distributes the traffic load on the gateway nodes. Through several simulations, we show that the proposed method can uniformly distribute the traf-fic load on gateway nodes, keeping the time complexity and suppressing an increase in the path length at most 15 % compared to the conventional shortest path routing.