Magic-angle twisted bilayer graphene exhibits intriguing quantum phase transitions triggered by enhanced electron-electron interactions when its flat bands are partially filled. However, the phases themselves and their connection to the putative non-trivial topology of the flat bands are largely unexplored. Here we report transport measurements revealing a succession of doping-induced Lifshitz transitions that are accompanied by van Hove singularities, which facilitate the emergence of correlation-induced gaps and topologically non-trivial subbands. In the presence of a magnetic field, well-quantized Hall plateaus at a filling of 1,2,3 carriers per moiré cell reveal the subband topology and signal the emergence of Chern insulators with Chern numbers, C = 3,2,1, respectively. Surprisingly, for magnetic fields exceeding 5 T we observe a van Hove singularity at a filling of 3.5, suggesting the possibility of a fractional Chern insulator. This van Hove singularity is accompanied by a crossover from low-temperature metallic, to high-temperature insulating behaviour, characteristic of entropically driven Pomeranchuk-like transitions.