The Central Molecular Zone (CMZ; inner ∼100pc) hosts some of the most dense and massive molecular clouds of the Milky Way. Studying these clouds can potentially lead to a better understanding of the dense clouds seen in the central starburst regions in nearby galaxies or in the early universe. The clouds share an unusual feature: they form stars at an unusually slow rate compared to other Milky Way clouds of similar mass and density. Here we use interferometer data from ALMA and the SMA to show that this reduced star formation rate is a consequence of the cloud density structure: CMZ clouds have unusually flat density slopes. The clouds do, for example, exceed the average density of the Orion A molecular cloud by an order of magnitude on spatial scales ∼5 pc, but the cores of CMZ clouds with ∼0.1 pc radius often have masses and densities lower than what is found in the Orion KL region. This relative absence of highest-density gas probably explains the suppression of star formation. The clouds are relatively turbulent, and ALMA observations of H 2 CO and SiO indicate that the turbulence is induced by high-velocity shocks. We speculate that these shocks might prevent the formation of high-mass cores.