This letter introduces a novel joint power-allocation (PA) and relay beamforming optimization (JPBO) algorithm for non-orthogonal multiple-access (NOMA) amplify-and-forward (AF) relay systems to achieve the maximum weighted sum-rate (WSR). In our proposed new framework, the WSR maximization problem is transformed into a tractable weighted-sum minimum mean-square-error (WSMMSE) problem. Then, the PA and the relay beamforming-matrix (BM) are alternately optimized by use of successive convex approximation (SCA) and Lagrange-multiplier (LM) algorithm. Simulation results demonstrate that our proposed new algorithm can significantly improve the WSR with a faster convergence than the existing method.