While the interaction of atoms in strong fields is well understood, the same cannot be said about molecules. We consider how dissociative ionization of molecules depends on the quality of the femtosecond laser pulses, in particular, the presence of third- and fourth-order dispersion. We find that high-order dispersion (HOD) unexpectedly results in order-of-magnitude enhanced ion yields, along with the factor of 3 greater kinetic energy release compared to transform limited pulses with equal peak intensities. The magnitude of these effects is not caused by increased pulse duration. We evaluate the role of pulse pedestals produced by HOD and other pulse shaping approaches, for a number of molecules including acetylene, methanol, methylene chloride, acetonitrile, toluene, and o-nitrotoluene, and discuss our findings in terms of processes such as prealignment, preionization, and bond softening. We conclude, based on the quasi-symmetric temporal dependence of the observed enhancements that cascade ionization is likely responsible for the large accumulation of charge prior to the ejection of energetic fragments along the laser polarization axis.