The knowledge of CSI at the BS (CSIT) is required to achieve the high spectrum efficiency promised by massive MIMO. In Frequency-Division Duplex (FDD) Massive MIMO systems, the CSIT is obtained via downlink channel estimation and uplink channel feedback, However, the acquisition of CSIT is a very challenging problem in practical FDD massive MIMO systems with a large number of antennas. Recently, compressive sensing has been applied to reduce pilot and CSIT feedback overheads in massive MIMO systems by exploiting the underlying channel sparsity. However, standard sparse recovery algorithms have stringent requirement on the channel sparsity level for robust channel recovery and this severely limits the operating regime of the solution. To overcome this issue, we propose a joint burst LASSO algorithm to exploit additional joint burst-sparse structure in multi-user (MU) massive MIMO channels. Simulations show that the joint burst LASSO algorithm can alleviate the stringent requirement on the sparsity level for robust channel recovery and substantially enhance the channel estimation performance over existing solutions.