This paper presents TurboNet, a novel model-driven deep learning (DL) architecture for turbo decoding that combines DL with the traditional max-log-maximum a posteriori (MAP) algorithm. To design TurboNet, we unfold the original iterative structure for turbo decoding and replace each iteration by a deep neural network (DNN) decoding unit. In particular, the DNN decoding unit is obtained by parameterizing the max-log-MAP algorithm rather than replace the whole decoder with a black box fully connected DNN architecture. With the proposed architecture, the parameters can be efficiently learned from training data, and thus TurboNet learns to appropriately use systematic and parity information to offer higher error correction capabilities and decrease computational complexity compared with existing methods. Furthermore, simulation results prove TurboNet's superiority in signal-to-noise ratio generalizations.