Background: The formation of superheavy elements (SHEs) by fusion of two massive nuclei is severely inhibited by the competing quasifission process. Lowexcitation energies favor SHE survival against fusion-fission competition. In "cold" fusion with spherical target nuclei near Pb-208, SHE yields are largest at beam energies significantly below the average capture barrier. In "hot" fusion with statically deformed actinide nuclei, this is not the case. Here the elongated deformation-aligned configurations in sub-barrier capture reactions inhibits fusion (formation of a compact compound nucleus), instead favoring rapid reseparation through quasifission. Purpose: To determine the probabilities of fast and slow quasifission in reactions with prolate statically deformed actinide nuclei, through measurement and quantitative analysis of the dependence of quasifission characteristics at beam energies spanning the average capture barrier energy. Methods: The Australian National University Heavy Ion Accelerator Facility and CUBE fission spectrometer have been used to measure fission and quasifission mass and angle distributions for reactions with projectiles from C to S, bombarding Th and U target nuclei. Results: Mass-asymmetric quasifission occurring on a fast time scale, associated with collisions with the tips of the prolate actinide nuclei, shows a rapid increase in probability with increasing projectile charge, the transition being centered around projectile atomic number ZP = 14. For mass-symmetric fission events, deviations of angular anisotropies from expectations for fusion fission, indicating a component of slower quasifission, suggest a similar transition, but centered around ZP similar to 8. Conclusions: Collisions with the tips of statically deformed prolate actinide nuclei show evidence for two distinct quasifission processes of different time scales. Their probabilities both increase rapidly with the projectile charge. The probability of fusion can be severely suppressed by these two quasifission processes, since the sub-barrier heavy element yield is likely to be determined by the product of the probabilities of surviving each quasifission process.