This paper considers a heterogeneous downlink cloud radio access network (C-RAN) where all the base stations (BSs) in the network are connected to a central processor (CP) via capacity-limited backhaul links. Under this model, we investigate the message-sharing transmission strategy where the CP shares each user's message with a fixed subset of BSs, which then serve the user through joint beamforming. In this setting, although the overall long-term average backhaul consumption is limited by the fixed cluster size, the instantaneous backhaul consumption at each BS may vary significantly depending on the data rates of the scheduled users at each time slot. To avoid such large fluctuations in backhaul consumption, this paper proposes a backhaul-aware multicell scheduling and beamforming strategy that explicitly accounts for backhaul consumption. Specifically, a beamforming design algorithm is proposed to maximize the network utility for a downlink C-RAN under both per-BS power constraints and per- BS backhaul constraints in each time slot. Although this problem has already been considered in our previous work, this present paper proposes a new beamforming design algorithm that not only has guaranteed convergence but also achieves better system performance. This paper also shows that the performance of the proposed algorithm can be further improved by iterating with an additional power minimization step.