Measured binary quasifission mass spectra in reactions with actinide nuclides show a large peak in yield near the doubly-magic 208Pb. This has generally been attributed to the enhanced binding energy of 208Pb causing a valley in the potential energy surface, attracting quasifission trajectories. To investigate this interpretation, binary quasifission mass spectra and cross-sections have been measured at near-barrier energies for reactions of 50Ti with actinide nuclides from 238U to 249Cf. Cross-sections have also been deduced for sequential fission (a projectile-like nucleus and two fragments from fission of the complementary target-like nucleus). Binary cross-sections fall from ∼70% of calculated capture cross-sections for 238U to only ∼40% for 249Cf, with a compensating increase in sequential fission cross-sections. The data are consistent with the 208Pb peak originating largely from sequential fission of heavier fragments produced in more mass-asymmetric primary quasifission events. These are increasingly suppressed as the heavy quasifission fragment mass increases above 208Pb. The important role of sequential fission calls for re-interpretation of quasifission characteristics and dynamics in superheavy element synthesis reactions.