Wireless mesh networks (WMNs) have emerged as a key technology for next generation wireless broadband networks showing rapid progress and inspiring numerous compelling applications. A WMN comprises of a set of mesh routers (MRs) and mesh clients (MCs), where MRs are connected to the Internet backbone through the Internet gateways (IGWs). The MCs are wireless devices and communicate among themselves over possibly multi-hop paths with or without the involvement of MRs. User privacy and security have been primary concerns in WMNs due to their peer-to-peer network topology, shared wireless medium, stringent resource constraints, and highly dynamic environment. Moreover, to support real-time applications, WMNs must also be equipped with robust, reliable and efficient routing protocols so as to minimize the end-to-end latency. Design of a secure and efficient routing protocol for WMNs, therefore, is of paramount importance. In this paper, we propose an efficient and reliable routing protocol that also provides user anonymity in WMNs. The protocol is based on an accurate estimation of the available bandwidth in the wireless links and a robust estimation of the end-to-end delay in a routing path, and minimization of control message overhead. The user anonymity, authentication and data privacy is achieved by application of a novel protocol that is based on Rivest's ring signature scheme. Simulations carried out on the proposed protocol demonstrate that it is more efficient than some of the existing routing protocols.