  In Bernal-stacked bilayer graphene (BBG), the Landau levels give rise to an
intimate connection between valley and layer degrees of freedom. Adding a
moir\'e superlattice potential enriches the BBG physics with the formation of
topological minibands - potentially leading to tunable exotic quantum
transport. Here, we present magnetotransport measurements of a high-quality
bilayer graphene-hexagonal boron nitride (hBN) heterostructure. The zero-degree
alignment generates a strong moir\'e superlattice potential for the electrons
in BBG and the resulting Landau fan diagram of longitudinal and Hall resistance
displays a Hofstadter butterfly pattern with a high level of detail. We
demonstrate that the intricate relationship between valley and layer degrees of
freedom controls the topology of moir\'e-induced bands, significantly
influencing the energetics of interacting quantum phases in the BBG
superlattice. We further observe signatures of field-induced correlated
insulators, helical edge states and clear quantizations of interaction-driven
topological quantum phases, such as symmetry broken Chern insulators.
