  We study wave-current interactions in two-dimensional water flows of constant
vorticity over a flat bed. For large-amplitude periodic traveling waves that
propagate at the water surface in the same direction as the underlying current
(downstream waves), we prove explicit uniform bounds for their amplitude. In
particular, our estimates show that the maximum amplitude of the waves becomes
vanishingly small as the vorticity increases without limit. We also prove that
the downstream waves on a global bifurcating branch are never overhanging, and
that their mass flux and Bernoulli constant are uniformly bounded.
