  Interactions among electrons and the topology of their energy bands can
create novel quantum phases of matter. Most topological electronic phases
appear in systems with weak electron-electron interactions. The instances where
topological phases emerge only as a result of strong interactions are rare, and
mostly limited to those realized in the presence of intense magnetic fields.
The discovery of flat electronic bands with topological character in
magic-angle twisted bilayer graphene (MATBG) has created a unique opportunity
to search for new strongly correlated topological phases. Here we introduce a
novel local spectroscopic technique using a scanning tunneling microscope (STM)
to detect a sequence of topological insulators in MATBG with Chern numbers C =
$\pm$ 1, $\pm$ 2, $\pm$ 3, which form near $\nu$ = $\pm$ 3, $\pm$ 2, $\pm$ 1
electrons per moir\'e unit cell respectively, and are stabilized by the
application of modest magnetic fields. One of the phases detected here (C = +1)
has been previously observed when the sublattice symmetry of MATBG was
intentionally broken by hexagonal boron nitride (hBN) substrates, with
interactions playing a secondary role. We demonstrate that strong
electron-electron interactions alone can produce not only the previously
observed phase, but also new and unexpected Chern insulating phases in MATBG.
The full sequence of phases we observed can be understood by postulating that
strong correlations favor 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 moir\'e systems
beyond those anticipated from weakly interacting models.
