Unmanned aerial vehicles (UAVs) have the advantages of high mobility, remarkable versatility, and on-demand deployment, which lead to their wide applications in various industries. In disasters or other emergency events, UAV-enabled relaying can provide a temporary wireless connection for the affected area. Thus, a UAV-enabled multi-link relaying system is considered in this article, where multiple sources communicate with their destinations via multiple UAV relays at the same time sharing the same spectrum. Considering the interference between any two source-UAV relay-destination links, we propose to maximize the minimum throughput of all links by jointly optimizing the UAV relays' three-dimension (3D) trajectories and the sources and UAV relays' transmit power levels. As the considered problem is non-convex, we propose an iterative algorithm to solve it by applying the block coordinate ascent technique, which optimizes the UAV trajectories and the transmit power alternately until reaching convergence. Numerical results show that the proposed algorithm significantly outperforms the benchmark algorithms in terms of all links' minimum throughput, and demonstrate the necessity of jointly optimizing the 3D UAV trajectories and transmit power for interference mitigation in a UAV-enabled multi-link relaying system.