The realization of robust intrinsic ferromagnetism in two-dimensional (2D) materials in conjunction with the intriguing quantum anomalous Hall (QAH) effect has provided a fertile ground for novel physics and for the next-generation spintronic and topological devices. On the basis of density functional theory (DFT), we predict that layered 5d transition-metal heavier halides (TMHs), such as ReX3 (X = Br, I), show intrinsic ferromagnetism with high spin polarization and high Curie temperatures. The outstanding dynamic and thermodynamic stability ensures their experimental feasibility. The strong spin-orbit coupling (SOC) of Re makes the electronic structure of the ReI3 monolayer topologically nontrivial with a large Chern number (C = -4). DFT+U calculations reveal that the 2D system undergoes a nontrivial to trivial transition with increasing on-site Hubbard Coulomb interaction U through the emergence of a Dirac cone. This transition is corroborated by the emergence of chiral edge states and the anomalous Hall conductivity. These findings not only demonstrate room-temperature ferromagnetism in atomically thin 5d TMHs, but also pave the way for the potential realization of the QAH effect with high Chern numbers in pristine 2D layers.