Bilayer graphene is a unique system where both the Fermi energy and the low-energy electron dispersion can be tuned. This is brought about by an interplay between trigonal warping and the band gap opened by a transverse electric field. Here, we drive the Lifshitz transition in bilayer graphene to experimentally controllable carrier densities by applying a large transverse electric field to a h-BN-encapsulated bilayer graphene structure. We perform magnetotransport measurements and investigate the different degeneracies in the Landau level spectrum. At low magnetic fields, the observation of filling factors -3 and -6 quantum Hall states reflects the existence of three maxima at the top of the valence-band dispersion. At high magnetic fields, all integer quantum Hall states are observed, indicating that deeper in the valence band the constant energy contours are singly connected. The fact that we observe ferromagnetic quantum Hall states at odd-integer filling factors testifies to the high quality of our sample. This enables us to identify several phase transitions between correlated quantum Hall states at intermediate magnetic fields, in agreement with the calculated evolution of the Landau level spectrum. The observed evolution of the degeneracies, therefore, reveals the presence of a Lifshitz transition in our system.