  New phases of matter can be stabilized by a combination of diverging
electronic density of states, strong interactions, and spin-orbit coupling.
Recent experiments in magic-angle twisted bilayer graphene (TBG) have uncovered
a wealth of novel phases as a result of interaction-driven spin-valley flavour
polarization. In this work, we explore correlated phases appearing due to the
combined effect of spin-orbit coupling-enhanced valley polarization and large
density of states below half filling ($\nu \lesssim 2$) of the moir\'e band in
a TBG coupled to tungsten diselenide. We observe anomalous Hall effect,
accompanied by a series of Lifshitz transitions, that are highly tunable with
carrier density and magnetic field. Strikingly, the magnetization shows an
abrupt sign change in the vicinity of half-filling, confirming its orbital
nature. The coercive fields reported are about an order of magnitude higher
than previous studies in graphene-based moir\'e systems, presumably aided by a
Stoner instability favoured by the van Hove singularities in the malleable
bands. While the Hall resistance is not quantized at zero magnetic fields,
indicative of a ground state with partial valley polarization, perfect
quantization and complete valley polarization are observed at finite fields.
Our findings illustrate that singularities in the flat bands in the presence of
spin-orbit coupling can stabilize ordered phases even at non-integer moir\'e
band fillings.
