  We perform an extensive exact diagonalization study of interaction driven
insulators in spin- and valley-polarized moir\'{e} flat bands of twisted
bilayer graphene aligned with its hexagonal boron nitride substrate. In
addition to previously reported fractional Chern insulator phases, we provide
compelling evidence for competing charge-density-wave phases at multiple
fractional fillings of a realistic single-band model. A thorough analysis at
different interlayer hopping parameters, motivated by experimental variability,
and the role of kinetic energy at various Coulomb interaction strengths
highlight the competition between these phases. The interplay of the
single-particle and the interaction induced hole dispersion with the inherent
Berry curvature of the Chern bands is intuitively understood to be the driving
mechanism for the ground-state selection. The resulting phase diagram features
remarkable agreement with experimental findings in a related moir\'{e}
heterostructure and affirms the relevance of our results beyond the scope of
graphene based materials.
