Turbo equalization is an iterative equalization and decoding technique that can achieve impressive performance gains for communication systems. In this letter, we investigate the turbo equalization method for the decoding of the Davey-MacKay (DM) construction over the IDS-AWGN channels, which indicates a cascaded insertion, deletion, substitution (IDS) channel and an additive white Gaussian noise (AWGN) channel. The inner decoder for the DM construction can be seen as an maximum a-posteriori (MAP) detector. It receives the beliefs generated by the outer LDPC decoder when turbo equalization is used. Two decoding schemes with different kinds of inner decoders, namely hard-input inner decoder and soft-input inner decoder, are investigated. Simulation results show that significant performance gains are obtained for both decoders with respect to the insertion/deletion probability at different SNR values.