Spectrum and energy efficiency with simultaneous wireless information and power transfer (SWIPT) to prolong the lifetime of power-constrained wireless devices in cooperative relaying nonorthogonal multiple access (CR-NOMA) has received great attention in the last decade. This paper investigates a two-way relay channel in a CR-NOMA system where two users exchange data with the assistance of a relay. Power-splitting relaying (PSR) and time-switching relaying (TSR) protocols are employed at the relay to harvest RF energy and process information from two users. We firstly derive the exact expressions of outage probability (OP) and system throughput (ST). The impacts of signal quality, energy coefficients, the distance of the nodes, and the data rate of two users on these performance metrics are then evaluated through several system settings to reflect practical network scenarios. It is shown that the OP and ST of the TSR are superior to that of the PSR protocol. Specifically, numerical results indicate that a higher throughput of up to 8% can be achieved with the TSR when compared to the PSR. It is further revealed that the OP and ST of the PSR are strongly affected by energy harvesting (EH) coefficients, while the performance obtained with the TSR is nearly independent of the EH capability at the relay.