Compressed sensing has been employed to reduce the pilot overhead for channel estimation in wireless communication systems. Particularly, structured turbo compressed sensing (STCS) provides a generic framework for structured sparse signal recovery with reduced computational complexity and storage requirement. In this paper, we consider the problem of massive multiple-input multiple-output (MIMO) orthogonal frequency division multiplexing (OFDM) channel estimation in a frequency division duplexing (FDD) downlink system. By exploiting the structured sparsity in the angle-frequency domain (AFD) and angle-delay domain (ADD) of the massive MIMO-OFDM channel, we represent the channel by using AFD and ADD probability models and design message-passing based channel estimators under the STCS framework. Several STCS-based algorithms are proposed for massive MIMO-OFDM channel estimation by exploiting the structured sparsity. We show that, compared with other existing algorithms, the proposed algorithms have a much faster convergence speed and achieve competitive error performance under a wide range of simulation settings.