We report on bandgap tuning in magnonic crystals made of nanometer-thick yttrium iron garnet (YIG) films with CoFeB-filled grooves via a variation of the groove depth, lattice constant, and film thickness. Using broadband spin-wave spectroscopy, we demonstrate bandgap widening in a 260-nm-thick YIG crystal when the grooves are deepened from half to full film thickness. Importantly, low-loss spin-wave transmission in the allowed bands of the magnonic crystal is almost unaffected by the patterning of fully discrete YIG stripes. Downscaling of the YIG film thickness to 35nm decreases the bandgap size through a flattening of the spin-wave dispersion relation. We show that a reduction in the lattice constant effectively compensates for this trend. Our experimental results are corroborated by micromagnetic simulations, providing relevant information for the design of ultrathin YIG-based magnonic crystals with optimized bandgaps and spin-wave transmission properties.