The emergence of high transition temperature (Tc) superconductivity in strongly correlated materials remains a major unsolved problem in physics. High-Tc materials, such as cuprates, are generally complex and not easily tunable, making theoretical modelling difficult. Although the Hubbard model--a simple theoretical model of interacting electrons on a lattice--is believed to capture the essential physics of high-Tc materials, obtaining accurate solutions of the model, especially in the relevant regime of moderate correlation, is challenging. The recent demonstration of robust superconductivity in moiré WSe2, whose low-energy electronic bands can be described by the Hubbard model and are highly tunable, presents a new platform for tackling the high-Tc problem. Here, we tune moiré WSe2 bilayers to the moderate correlation regime through the twist angle and map the phase diagram around one hole per moiré unit cell (v = 1) by electrostatic gating and electrical transport and magneto-optical measurements. We observe a range of high-Tc phenomenology, including an antiferromagnetic insulator at v = 1, superconducting domes upon electron and hole doping, and unusual metallic states at elevated temperatures including strange metallicity. The highest Tc occurs adjacent to the Mott transition, reaching about 6% of the effective Fermi temperature. Our results establish a new material system based on transition metal dichalcogenide (TMD) moiré superlattices that can be used to study high-Tc superconductivity in a highly controllable manner and beyond.