The demand of massive access to the same multimedia content at the same time is one major challenge for next-generation cellular networks in densely-packed urban areas. The content-aware multicast transmission strategies provide promising solutions to such use cases involving many mobile users trying to fetch the same data. In this work, we consider multigroup multicast transmission with a common message (e.g., multimedia content), in which different multicast groups are interested along with their private multicast messages. We further assume that a relay helps the cellular base station (BS) disseminate multicast content to the users experiencing high path loss and/or blockage. We propose superposition and concatenated coding schemes, denoted by SC and CC, respectively, to transmit the common and private multicast messages. In order to maximize max-min fair (MMF) rates, we design a novel low-complexity alternating-optimization algorithm to compute transmit and relay precoders. We also propose rate-splitting (RS) alternatives of SC and CC schemes together with an iterative algorithm to derive dedicated transmit and relay precoders so as to maximize MMF rates. We rigorously evaluate the performance of the proposed transmission schemes and precoders, and verify the superiority of RS-based schemes in overloaded scenarios without any saturation with increasing signal-to-noise ratio.