  We develop a new approach to $L^{\infty}$-a priori estimates for degenerate
complex Monge-Amp\`ere equations on complex manifolds. It only relies on
compactness and envelopes properties of quasi-plurisubharmonic functions. In a
prequel \cite{GL21a} we have shown how this method allows one to obtain new and
efficient proofs of several fundamental results in K\"ahler geometry. In
\cite{GL21b} we have studied the behavior of Monge-Amp\`ere volumes on
hermitian manifolds. We extend here the techniques of \cite{GL21a} to the
hermitian setting and use the bounds established in \cite{GL21b}, producing new
relative a priori estimates, as well as several existence results for
degenerate complex Monge-Amp\`ere equations on compact hermitian manifolds.
