This paper presents an irregular distributed space-time (Ir-DST) coding scheme designed for near-capacity cooperative communications where the system's effective throughput is also maximized with the aid of a joint source-and-relay mode design procedure. At the source node, a serially concatenated scheme comprising an IRregular Convolutional Code (IRCC), a recursive unity-rate code (URC), and a space-time block code (STBC) was designed for the sake of approaching the corresponding source-to-relay link capacity, where the IRCC was optimized with the aid of EXtrinsic Information Transfer (EXIT) charts. At the relay node, another IRCC is serially concatenated with an identical STBC. Before transmitting the relayed information, the relay's IRCC is reoptimized based on EXIT chart analysis for the sake of approaching the relay channel's capacity and to maximize the relay's coding rate, which results in a maximized effective throughput. We will demonstrate that the topology of the Ir-DST system coincides with that of a distributed turbo code (DTC). At the destination node, a novel three-stage iterative decoding scheme is constructed to achieve decoding convergence to an infinitesimally low bit-error ratio (BER). Finally, our numerical results show the proficiency of our joint source-and-relay mode design procedure, demonstrating that the proposed Ir-DST coding scheme is capable of near-capacity cooperative communications and of maximizing the effective throughput.