Cloud-radio access networks (C-RAN) help overcoming the scarcity of radio resources by enabling dense deployment of base-stations (BSs), and connecting them to a central-processor (CP). This paper considers the downlink of a C-RAN, where the cloud is connected to the BSs via limited-capacity backhaul links. The paper proposes splitting the message of each user into two parts, a private part decodable at the intended user only, and a common part which can be decoded at a subset of users, as a means to enable large-scale interference management in CRAN. To this end, the paper optimizes a transmission scheme that combines rate splitting (RS), common message decoding (CMD), clustering and coordinated beamforming. The paper focuses on maximizing the weighted sum-rate subject to per-BS backhaul capacity and transmit power constraints, so as to jointly determine the RS-CMD mode of transmission, the cluster of BSs serving private and common messages of each user, and the associated beamforming vectors of each user private and common messages. The paper proposes solving such a complicated non-convex optimization problem using $l_0$-norm relaxation techniques, followed by inner-convex approximations (ICA), so as to achieve stationary solutions to the relaxed non-convex problem. Numerical results show that the proposed method provides significant performance gain as compared to conventional interference mitigation techniques in CRAN which treat interference as noise (TIN).