The Taffy system (UGC 12914/15) consists of two massive spiral galaxies which had a head-on collision about 20Myr ago. New sensitive, high-resolution CO(1-0) observations of the Taffy system with the IRAM PdBI are presented. About 25% of the total interferometric CO luminosity stems from the bridge region. Assuming a Galactic N(H2)/ICO conversion factor for the galactic disks and a third of this value for the bridge gas, about 10% of the molecular gas mass is located in the bridge region. The giant HII region close to UGC 12915 is located at the northern edge of the high-surface brightness giant molecular cloud association (GMA), which has the highest velocity dispersion among the bridge GMAs. The bridge GMAs are clearly not virialized because of their high velocity dispersion. Three dynamical models are presented and while no single model reproduces all of the observed features, they are all present in at least one of the models. Most of the bridge gas detected in CO does not form stars. We suggest that turbulent adiabatic compression is responsible for the exceptionally high velocity dispersion of the molecular ISM and the suppression of star formation in the Taffy bridge. In this scenario the turbulent velocity dispersion of the largest eddies and turbulent substructures/clouds increase such that giant molecular clouds are no longer in global virial equilibrium. The increase of the virial parameter leads to a decrease of the star formation efficiency. Most of the low-surface density, CO-emitting gas will disperse without forming stars but some of the high-density gas will probably collapse and form dense star clusters, such as the luminous HII region close to UGC 12915. We suggest that globular clusters and super star clusters formed and still form through the gravitational collapse of gas previously compressed by turbulent adiabatic compression during galaxy interactions.