  A controlled twist between different underlying lattices allows one to
interpolate, under a unified framework, across ordered and (quasi-)disordered
matter while drastically changing quantum transport properties. Here, we use
quantum Monte Carlo simulations to determine the unique phase diagrams of
strongly-correlated ultracold bosons in twisted optical potentials. We show
that at commensurate twisting angles, spectral gaps govern the formation of
insulating patterns, separated by thin superfluid domains. The latter form weak
superfluids, which are very sensitive to thermal fluctuations, but can be
stabilized under appropriate parameter control. In contrast, slightly changing
the twisting angle to a incommensurate value destroys most spectral gaps,
leaving behind a prominent Bose glass phase. Our results are directly
applicable to current generation experiments that quantum simulate moir\'e
physics.
