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 at half-integer band-fillings of $ u = 0.5$ and $3.5$ at $Bapprox 0$ in a TBG proximitized by a layer of tungsten diselenide (WSe$_2$). The magnetotransport data enables us to deduce features in the underlying band structure consistent with a spontaneously broken translational symmetry supercell with twice the area of the original TBG moir{e} cell. A series of Lifshitz transitions due to the changes in the topology of the Fermi surface implies the evolution of van Hove singularities (VHS) of the diverging density of states at a discrete set of partial fillings of flat bands. Further, we observe reset of charge carriers at $ u = 2, 3$. In addition to magnetotransport, we employ thermoelectricity as a tool to probe the system at $B=0$. Band structure calculations for a TBG moir{e} pattern, together with a commensurate density wave potential and spin-orbit coupling (SOC) terms, allow to obtain degeneracy-lifted, zone-folded moir{e} bands with spin-valley isospin ordering anisotropy that describe the states at half-integer fillings observed experimentally. Our results suggest the emergence of a spin-charge density wave ground state in TBG in the zero $B-$ field limit.
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