The triple-alpha reaction rate in proton-rich core-collapse supernovae is found to be enhanced at high nucleon densities, suppressing the formation of proton-rich nuclei from gallium to cadmium. The rate of the triple-alpha reaction that forms C-12 affects(1,2) the synthesis of heavy elements in the Ga-Cd range in proton-rich neutrino-driven outflows of core-collapse supernovae(3-5). Initially, these outflows contain only protons and neutrons; these later combine to form alpha particles, then C-12 nuclei via the triple-alpha reaction, and eventually heavier nuclei as the material expands and cools. Previous experimental work(6,7) demonstrated that despite the high temperatures encountered in these environments, the reaction is dominated by the well characterized Hoyle state resonance in C-12 nuclei. At sufficiently high nucleon densities, however, proton- and neutron-scattering processes may alter the effective width of the Hoyle state(8,9). This raises the questions of what the reaction rate in supernova outflows is, and how changes affect nucleosynthesis predictions. Here we report that in proton-rich core-collapse supernova outflows, these hitherto neglected processes enhance the triple-alpha reaction rate by up to an order of magnitude. The larger reaction rate suppresses the production of heavy proton-rich isotopes that are formed by the nu p process(3-5) (where nu is the neutrino and p is the proton) in the innermost ejected material of supernovae(10-13). Previous work on the rate enhancement mechanism(9) did not anticipate the importance of this enhancement for proton-rich nucleosynthesis. Because the in-medium contribution to the triple-alpha reaction rate must be present at high densities, this effect needs to be included in supernova nucleosynthesis models. This enhancement also differs from earlier sensitivity studies that explored variations of the unenhanced rate by a constant factor(1,2), because the enhancement depends on the evolving thermodynamic conditions. The resulting suppression of heavy-element nucleosynthesis for realistic conditions casts doubt on the nu p process being the explanation for the anomalously high abundances of Mo-92,Mo-94 and Ru-96,Ru-98 isotopes in the Solar System(1,3,14) and for the signatures of early Universe element synthesis in the Ga-Cd range found in the spectra of ancient metal-poor stars(15-20).