Using the TNG100 (100 Mpc)^3 simulation of the IllustrisTNG project, we demonstrate a strong connection between the onset of star formation quenching and the stellar size of galaxies. We do so by tracking the evolutionary history of extended and normal-size galaxies selected at z=2 with log(M_star) = 10.2 - 11 and stellar-half-mass-radii above and within 1-sigma of the stellar size--stellar mass relation, respectively. We match the stellar mass and star formation rate distributions of the two populations. By z=1, only 36% of the extended massive galaxies have quenched, in contrast to a quenched fraction of 69% for the normal-size massive galaxies. We find that normal-size massive galaxies build up their central stellar mass without a significant increase in their stellar size between z=2-4, whereas the stellar size of the extended massive galaxies almost doubles in the same time. In IllustrisTNG, lower black hole masses and weaker kinetic-mode feedback appears to be responsible for the delayed quenching of star formation in the extended massive galaxies. We show that relatively gas-poor mergers may be responsible for the lower central stellar density and weaker supermassive black hole feedback in the extended massive galaxies.