We use Keck/NIRSPEC to survey a sample of of young (<1 Gyr), short-period mini-Neptunes orbiting nearby K dwarfs to measure their mass loss via the metastable helium line. We detect helium absorption from all four of the targets in our initial sample. The first detection, around TOI 560b, was announced in a previous paper. We now announce three additional detections around TOI 1430.01, 2076b, and 1683.01. All four planets show an average in-transit excess absorption of 0.7%–1.0%. However, the outflows differ in their kinematic properties. Object TOI 1430b exhibits preingress absorption, while TOI 2076b’s outflow is exceptionally optically thick and shows significant postegress absorption. For all four planets, the width of the measured helium absorption signal is consistent with expectations for a photoevaporative outflow (10–30 km s−1, 5000–10,000 K). Unless broadening mechanisms other than thermal velocity and the bulk outflow velocity are significant, our observations disfavor core-powered mass-loss models, which predict much slower (1–3 km s−1) outflows. We utilize both an isothermal Parker wind model and an order-of-magnitude method to estimate the mass-loss timescale and obtain ∼a few hundred megayears for each planet. We conclude that many, if not all, of these planets will lose their hydrogen-rich envelopes and become super-Earths. Our results demonstrate that most mini-Neptunes orbiting Sun-like stars have primordial atmospheres, and that photoevaporation is an efficient mechanism for stripping these atmospheres and transforming these planets into super-Earths.