  The dominance of Coulomb interactions over kinetic energy of electrons in
narrow, non-trivial moir\'{e} bands of magic-angle twisted bilayer graphene
(TBG) gives rise to a variety of correlated phases such as correlated
insulators, superconductivity, orbital ferromagnetism, Chern insulators and
nematicity. Most of these phases occur at or near an integer number of carriers
per moir\'{e} unit cell. Experimental demonstration of ordered states at
fractional moir\'{e} band-fillings at zero applied magnetic field $B$, is a
challenging pursuit. In this letter, we report the observation of states near
half-integer band-fillings of $\nu\approx 0.5$ and $\pm3.5$ at $B\approx 0$ in
TBG proximitized by tungsten diselenide (WSe$_2$) through magnetotransport and
thermoelectricity measurements. A series of Lifshitz transitions due to the
changes in the topology of the Fermi surface implies the evolution of van Hove
singularities (VHSs) of the diverging density of states (DOS) at a discrete set
of partial fillings of flat bands. Furthermore, at a band filling of
$\nu\approx-0.5$, a symmetry-broken Chern insulator emerges at high $B$,
compatible with the band structure calculations within a translational
symmetry-broken supercell with twice the area of the original TBG moir\'{e}
cell. Our results are consistent with a spin/charge density wave ground state
in TBG in the zero $B$-field limit.
