Strong electron-electron interactions in magic-angle twisted bilayer graphene can fundamentally change the topology of the system's flat bands, producing a hierarchy of strongly correlated topological insulators in modest magnetic fields. Interactions between electrons and the topology of their energy bands can create unusual quantum phases of matter. Most topological electronic phases appear in systems with weak electron-electron interactions. The instances in which topological phases emerge only as a result of strong interactions are rare and mostly limited to those realized in intense magnetic fields(1). The discovery of flat electronic bands with topological character in magic-angle twisted bilayer graphene (MATBG) has created a unique opportunity to search for strongly correlated topological phases(2-9). Here we introduce a local spectroscopic technique using a scanning tunnelling microscope to detect a sequence of topological insulators in MATBG with Chern numbers C = +/- 1, +/- 2 and +/- 3, which form near filling factors of +/- 3, +/- 2 and +/- 1 electrons per moire unit cell, respectively, and are stabilized by modest magnetic fields. One of the phases detected here (C = +1) was previously observed when the sublattice symmetry of MATBG was intentionally broken by a hexagonal boron nitride substrate, with interactions having a secondary role(9). We demonstrate that strong electron-electron interactions alone can produce not only the previously observed phase, but also other unexpected Chern insulating phases in MATBG. The full sequence of phases that we observe can be understood by postulating that strong correlations favour breaking time-reversal symmetry to form Chern insulators that are stabilized by weak magnetic fields. Our findings illustrate that many-body correlations can create topological phases in moire systems beyond those anticipated from weakly interacting models.