Abstract We present Atacama Large Millimeter/submillimeter Array observations at a spatial resolution of 0.″2 (60 pc) of CO emission from the Taffy galaxies (UGC 12914/5). The observations are compared with narrowband Paα, mid-IR, radio continuum and X-ray imaging, plus optical spectroscopy. The galaxies have undergone a recent head-on collision, creating a massive gaseous bridge that is known to be highly turbulent. The bridge contains a complex web of narrow molecular filaments and clumps. The majority of the filaments are devoid of star formation, and fall significantly below the Kennicutt–Schmidt relationship for normal galaxies, especially for the numerous regions undetected in Paα emission. Within the loosely connected filaments and clumps of gas we find regions of high velocity dispersion that appear gravitationally unbound for a wide range of likely values of X _CO. Like the “Firecracker” region in the Antennae system, they would require extremely high external dynamical or thermal pressure to stop them dissipating rapidly on short crossing timescales of 2–5 Myr. We suggest that the clouds may be transient structures within a highly turbulent multiphase medium that is strongly suppressing star formation. Despite the overall turbulence in the system, stars seem to have formed in compact hotspots within a kiloparsec-sized extragalactic H ii region, where the molecular gas has a lower velocity dispersion than elsewhere, and shows evidence for a collision with an ionized gas cloud. Like the shocked gas in the Stephan’s Quintet group, the conditions in the Taffy bridge shows how difficult it is to form stars within a turbulent, multiphase, gas.