Trans-Neptunian objects found in the first four years of the Dark Energy Survey

We present a catalog of 316 trans-Neptunian bodies detected by the Dark Energy Survey (DES). These objects include 245 discoveries by DES (139 not previously published) detected in $\approx 60,000$ exposures from the first four seasons of the survey ("Y4" data). The survey covers a contiguous 5000 deg$^2$ of the southern sky in the $grizY$ optical/NIR filter set, with a typical TNO in this part of the sky being targeted by $25-30$ Y4 exposures. We describe the processes for detection of transient sources and the linkage into TNO orbits, which are made challenging by the absence of the few-hour repeat observations employed by TNO-optimized surveys. We also describe the procedures for determining detection efficiencies vs. magnitude and estimating rates of false-positive linkages. This work presents all TNOs which were detected on $\ge 6$ unique nights in the Y4 data and pass a "sub-threshold confirmation" test wherein we demand the the object be detectable in a stack of the individual images in which the orbit indicates an object should be present, but was not detected. This eliminates false positives and yields TNO detections complete to $r\lesssim 23.3$ mag with virtually no dependence on orbital properties for bound TNOs at distance $30\,{\rm AU} 0.3$ mag more depth, and arcs of $>4$ years for nearly all detections.