In this paper, a novel approach is proposed to improve the minimum signal-to-interference-plus-noise-ratio (SINR) among users in non-orthogonal multi-user wireless relay networks, by optimizing the placement of unmanned aerial vehicle (UAV) relays, relay beamforming, and receive combining. The design is separated into two problems: beamforming-aware UAV placement optimization and transceiver design for minimum SINR maximization. A significant challenge in beamforming-aware UAV placement optimization is the lack of instantaneous channel state information (CSI) prior to deploying UAV relays, making it difficult to derive the beamforming SINR in non-orthogonal multi-user transmission. To address this issue, an approximation of the expected beamforming SINR is derived using the narrow beam property of a massive MIMO base station. Based on this, a UAV placement algorithm is proposed to provide UAV positions that improve the minimum expected beamforming SINR among users, using a difference-of-convex framework. Subsequently, after deploying the UAV relays to the optimized positions, and with estimated CSI available, a joint relay beamforming and receive combining (JRBC) algorithm is proposed to optimize the transceiver to improve the minimum beamforming SINR among users, using a block-coordinate descent approach. Numerical results show that the UAV placement algorithm combined with the JRBC algorithm provides a 4.6 dB SINR improvement over state-of-the-art schemes.