We report the results of optical follow-up observations of 29 gravitational-wave (GW) triggers during the first half of the LIGO-Virgo Collaboration (LVC) O3 run with the Gravitational-wave Optical Transient Observer (GOTO) in its prototype 4-telescope configuration (GOTO-4). While no viable electromagnetic (EM) counterpart candidate was identified, we estimate our 3D (volumetric) coverage using test light curves of on- and off-axis gamma-ray bursts and kilonovae. In cases where the source region was observable immediately, GOTO-4 was able to respond to a GW alert in less than a minute. The average time of first observation was 8.79 h after receiving an alert (9.90 h after trigger). A mean of 732.3 square degrees were tiled per event, representing on average 45.3 per cent of the LVC probability map, or 70.3 per cent of the observable probability. This coverage will further improve as the facility scales up alongside the localization performance of the evolving GW detector network. Even in its 4-telescope prototype configuration, GOTO is capable of detecting AT2017gfo-like kilonovae beyond 200 Mpc in favourable observing conditions. We cannot currently place meaningful EM limits on the population of distant ( $\hat{D}_L = 1.3$ Gpc) binary black hole mergers because our test models are too faint to recover at this distance. However, as GOTO is upgraded towards its full 32-telescope, 2 node (La Palma & Australia) configuration, it is expected to be sufficiently sensitive to cover the predicted O4 binary neutron star merger volume, and will be able to respond to both northern and southern triggers.