Within the first few hundreds of millions of years, many physical processes sculpt the eventual properties of young planets. NASA's Transiting Exoplanet Survey Satellite (TESS) mission has surveyed young stellar associations across the entire sky for transiting planets, providing glimpses into the various stages of planetary evolution. Using our own detection pipeline, we search a magnitude-limited sample of 7219 young stars (<~200Myr) observed in the first 4 yr of TESS for small (2-8R{Earth}), short period (1.6-20days) transiting planets. The completeness of our survey is characterized by a series of injection and recovery simulations. Our analysis of TESS 2minute cadence and Full Frame Image (FFI) light curves recover all known TESS Objects of Interest (TOIs), as well as four new planet candidates not previously identified as TOIs. We derive an occurrence rate of 35_-10_^+13^% for mini-Neptunes and 27_-8_^+10^% for super-Neptunes from the 2 minute cadence data, and 22_-6.8_^+8.6^% for mini-Neptunes and 13_-4.9_^+3.9^% for super-Neptunes from the FFI data. To independently validate our results, we compare our survey yield with the predicted planet yield assuming Kepler planet statistics. We consistently find a mild increase in the occurrence of super-Neptunes and a significant increase in the occurrence of Neptune-sized planets with orbital periods of 6.2-12days when compared to their mature counterparts. The young planet distribution from our study is most consistent with evolution models describing the early contraction of hydrogen-dominated atmospheres undergoing atmospheric escape and inconsistent with heavier atmosphere models offering only mild radial contraction early on.