We present a study of laser-driven ion acceleration with micrometer and submicrometer thick plastic targets. Using laser pulses with high temporal contrast and an intensity of the order of 10^{20} W/cm^{2} we observe proton beams with cutoff energies in excess of 85 MeV and particle numbers of 10^{9} in an energy bin of 1 MeV around this maximum. We show that applying the target normal sheath acceleration mechanism with submicrometer thick targets is a very robust way to achieve such high ion energies and particle fluxes. Our results are backed with 2D particle in cell simulations furthermore predicting cutoff energies above 200 MeV for acceleration based on relativistic transparency. This predicted regime can be probed after a few technically feasible adjustments of the laser and target parameters.