Accurate channel estimation with small pilot overhead is vital to improve the capacity and reliability of massive MIMO systems. Recently, compressed sensing has been applied to reduce the pilot overhead in such systems by exploiting the underlying structured channel sparsity. In this paper, we propose a structured turbo compressed sensing (Turbo-CS) framework for the design and analysis of structured sparse channel estimation algorithms. In this framework, a Markov prior is used to model the structured sparsity in massive MIMO channels. Then we extend the Turbo-CS algorithm for independent and identically distributed priors to propose a structured Turbo-CS algorithm to solve the resulting sparse channel estimation problem with the Markov chain prior. We also accurately characterize the performance of the algorithm using state evolution. As compared to the existing algorithms, both the state evolution analysis and simulations show that the structured Turbo-CS algorithm can substantially enhance the channel estimation performance.