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This paper investigates a cooperative short-packet non-orthogonal multiple access-based relay-sharing system, in which a common relay node assists the communication of two source-destination pairs. Aiming at reducing relay complexity design and transmission latency, a novel relaying protocol, which is composed of fewer encoders and decoders than decode-and-forward (DF) relaying protocol, is proposed. Accordingly, approximate and asymptotic expressions for the end-to-end average block error rates (BLERs) are derived, and numerical results show the superior BLER performance of the proposed protocol over the conventional DF, amplify-and-forward, and orthogonal multiple access ones.