This paper proposes a new communication protocol for a cooperative non-orthogonal multiple access (NOMA) system. In this system, based on users' channel conditions, each two NOMA users are paired to reduce system complexity. In this concern, the user with a better channel condition decodes and then forwards messages received from the source to the user with a worse channel condition. In particular, the direct link between the paired users is assumed to be unavailable due to the weak transmission conditions. To overcome this issue, we propose a new cooperative NOMA protocol in which an amplify-and-forward (AF) relay is employed to help the user-to-user communications. To evaluate the proposed protocol, the exact closed-form expressions of outage probability (OP) at the two paired users are derived. Based on the analysis of the OP, we further examine the system throughput in a delay-sensitive transmission mode. Finally, our analytical results verified by Monte-Carlo simulation show that the proposed protocol is efficient in enhancing the performance of NOMA system when the user-to-user communications is limited.