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In recent years, wireless mesh network (WMN) is a compelling topic to many network researchers due to its low cost in deployment, simplicity in installation and robustness in operation. However, existing routing protocols designed for MANET can not work efficiently in WMN because backbone in WMN formed by Mesh Router has very low mobility and are not put under power and memory constraint. In addition, they just concentrate on finding single path from the source to the destination while the characteristics of robustness of WMN require that every intermediate Mesh Router should establish several paths from itself to its desired destination. In our paper, we design a routing protocol that permits each MR to quickly discover multiple paths based on Hop Count metric to the Internet Gateways. Unfortunately, Hop Count does not take packet loss or bandwidth into account. It results in low throughput of a flow that follows the shortest path. A bandwidth estimation technique has been proposed to apply at each Mesh Router to allow it to predict congestion risk over its connected links and to select high available bandwidth link for forwarding packets. A detailed performance evaluation shows that the throughput increases greatly compared to pure AODV and AOMDV in high-loaded traffic scenario.