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Since the introduction of turbo-code-aided Hybrid Automatic Repeat reQuest (HARQ) schemes, their complexity reduction has drawn research attention. In our previous work, we proposed an Early Stopping (ES) strategy for a turbo HARQ scheme, which results in beneficial complexity reduction while maintaining a high throughput. However, this scheme was designed to strike a compromise across the full range of channel conditions. As a further advance, in this paper, we propose a new Deferred Iterations (DI) strategy, which is specifically designed to take into account the prevalent channel conditions, as characterized by the Extrinsic Information Transfer (EXIT) chart tunnel opening. More specifically, the DI strategy delays the commencement of the turbo-decoding until an open EXIT chart tunnel appears. Our simulation results demonstrate that the complexity of the proposed DI-aided turbo HARQ schemes is reduced by up to 50%, which is achieved without compromising the packet loss ratio (PLR) or throughput.