Carbon and oxygen are key tracers of the Galactic chemical evolution; in particular, a reported upturn in [C/O] towards decreasing [O/H] in metal-poor halo stars could be a signature of nucleosynthesis by massive Population III stars. We reanalyse carbon, oxygen, and iron abundances in 39 metal-poor turn-off stars. For the first time, we take into account 3D hydrodynamic effects together with departures from local thermodynamic equilibrium (LTE) when determining both the stellar parameters and the elemental abundances, by deriving effective temperatures from 3D non-LTE H{beta} profiles, surface gravities from Gaia parallaxes, iron abundances from 3D LTE FeII equivalent widths, and carbon and oxygen abundances from 3D non-LTE CI and OI equivalent widths. We find that [C/Fe] stays flat with [Fe/H], whereas [O/Fe] increases linearly up to 0.75dex with decreasing [Fe/H] down to -3.0dex. Therefore [C/O] monotonically decreases towards decreasing [C/H], in contrast to previous findings, mainly because the non-LTE effects for OI at low [Fe/H] are weaker with our improved calculations.