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, and evaluates rate-splitting (RS) and common-message decoding techniques, as a means to enable large-scale interference management. To this end, the paper proposes splitting the message of each user at the CP into a private part decodable at one user, and a common part decodable at a subset of users for the sole purpose of interference mitigation. The paper then focuses on maximizing the weighted sum-rate subject to backhaul capacity and transmission power constraints, so as to determine the RS mode of each user, and the associated beamforming vectors. The paper proposes solving such a complicated non-convex optimization problem using an inner-convex approximation approach, which guarantees achieving a stationary solution to the problem. Numerical results show that the proposed method provides significant gain compared to classical interference mitigation techniques that do not rely on RS and common message decoding.