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We report the results of optical follow-up observations of 29 gravitational-wave 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 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 $8.79$ hours after receiving an alert ($9.90$ hours 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 localisation performance of the evolving gravitational-wave 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 electromagnetic 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.