The radius and surface composition of an exploding massive star, as well as the explosion energy per unit mass, can be measured using early ultraviolet (UV) observations of core-collapse supernovae (CC SNe). We present the results from a simultaneous Galaxy Evolution Explorer (GALEX) and Palomar Transient Factory (PTF) search for early UV emission from SNe. We analyze five CC SNe for which we obtained near-UV (NUV) measurements before the first ground-based R-band detection. We introduce SOPRANOS, a new maximum likelihood fitting tool for models with variable temporal validity windows, and use it to fit the Sapir & Waxman (2017ApJ...838..130S) shock-cooling model to the data. We report four Type II SNe with progenitor radii in the range of R_*_~600-1100R_{sun}_ and a shock velocity parameter in the range of v_s*_~2700-6000km/s (E/M~2-8x10^50^erg/M_{sun}_) and one Type IIb SN with R_*_~210R_{sun}_ and v_s*_~11000km/s (E/M~1.8x10^51^erg/M_{sun}_). Our pilot GALEX/PTF project thus suggests that a dedicated, systematic SN survey in the NUV band, such as the wide-field UV explorer ULTRASAT mission, is a compelling method to study the properties of SN progenitors and SN energetics.