  Diameter estimates for K\"ahler metrics are established which require only an
entropy bound and no lower bound on the Ricci curvature. The proof builds on
recent PDE techniques for $L^\infty$ estimates for the Monge-Amp\`ere equation,
with a key improvement allowing degeneracies of the volume form of codimension
strictly greater than one. As a consequence, diameter bounds are obtained for
long-time solutions of the K\"ahler-Ricci flow and finite-time solutions when
the limiting class is big, as well as for special fibrations of Calabi-Yau
manifolds.
