In this article, a bidirectional relaying (BR) non-orthogonal multiple access (NOMA) with simultaneous wireless information and power transfer (SWIPT) termed as BR NOMA-SWIPT is proposed and investigated for the Internet of Things (IoT) relay networks. Here, multiple NOMA users in one group can communicate or exchange information with multiple NOMA users in another group through a common energy harvesting (EH) based relay. The EH based relay exploits the radio frequency (RF) energy supplied by the two NOMA user groups to recharge itself, and then it exchanges the information between them. Specifically, the two groups of NOMA users transmit the information intended for the exchange to the relay node using the uplink NOMA protocol. The relay node first harvests the RF energy through the signals of the two group of NOMA users, and then it carries out the exchange of information between two NOMA user groups by using the downlink NOMA protocol. To the best of our knowledge, there is no existing study or research on BR NOMA with SWIPT. Therefore, in this article, we examine in detail, the performance of BR NOMA-SWIPT enabled IoT relay networks. Specifically, we study a popular EH time switching (TS) relaying architecture with BR and NOMA. We also study the effect of both perfect successive interference cancellation (pSIC) and imperfect SIC (ipSIC) on the proposed BR NOMA-SWIPT system. Analytical expressions for the outage probability and ergodic capacity are mathematically derived. The analytical results of our proposed system model are validated by the simulation results, and representative performance comparisons are presented thoroughly, which not only provides practical insights into the effect of different system parameters on the overall network performance, but it also demonstrates that our proposed BR NOMA-SWIPT can attain significant throughput and capacity gains as compared to conventional BR multiple access schemes.