Detection of transiting exoplanets around young stars is more difficult than for older systems owing to increased stellar variability. Nine young open cluster planets have been found in the K2 data, but no single analysis pipeline identified all planets. We have developed a transit search pipeline for young stars that uses a transit-shaped notch and quadratic continuum in a 12 or 24 hr window to fit both the stellar variability and the presence of a transit. In addition, for the most rapid rotators (P_rot_<2 days) we model the variability using a linear combination of observed rotations of each star. To maximally exploit our new pipeline, we update the membership for four stellar populations observed by K2 (Upper Scorpius, Pleiades, Hyades, Praesepe) and conduct a uniform search of the members. We identify all known transiting exoplanets in the clusters, 17 eclipsing binaries, one transiting planet candidate orbiting a potential Pleiades member, and three orbiting unlikely members of the young clusters. Limited injection recovery testing on the known planet hosts indicates that for the older Praesepe systems we are sensitive to additional exoplanets as small as 1-2 R_{Earth}_, and for the larger Upper Scorpius planet host (K2-33) our pipeline is sensitive to ~4 R_{Earth}_ transiting planets. The lack of detected multiple systems in the young clusters is consistent with the expected frequency from the original Kepler sample, within our detection limits. With a robust pipeline that detects all known planets in the young clusters, occurrence rate testing at young ages is now possible.