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This paper investigates a cooperative non-orthogonal multiple access (C-NOMA) network consisting of a base station (BS), a far user (FU) and a full-duplex (FD) near user (NU). The NU does not use its own battery energy to relay to FU, and harvests energy using SWIPT principles from the BS. In this paper we consider a more practical nonlinear energy harvesting model for the first time in such a framework. We show that use of the idealized linear EH model leads to gross over-estimation of FU performance. Considering the time-splitting (TS) energy harvesting (EH) protocol, and modelling the self-interference at the FD NU, expressions are derived for the FU and NU outage probabilities in closed-form. Further, we show that by optimal selection of the TS parameter performance of FU can be further enhanced. Computer simulations confirm the accuracy of the derived analytical expressions.