We propose a detection strategy for the two-dimensional magnetic recording channel with multiple read heads, in which intertrack interference (ITI) is mitigated by a combination of linear combining and soft interference cancellation. The proposed strategy reduces the detection problem to a series of traditional one-dimensional detection problems, so that the existing iterative detection strategies based on turbo equalization may be leveraged. The performance of the proposed detector is compared with a previously reported detector, in which the ITI is suppressed linearly via a multiple-input equalizer. Numerical results demonstrate that the proposed detector outperforms the linear detector regardless of whether or not media noise is dominant, and further that the proposed detector is able to recover data from more tracks than is otherwise possible.