Unlike coffee and cream that homogenize when stirred, growing micro-organisms (e.g., bacteria, baker's yeast) can actively kill each other and avoid mixing. How do such antagonistic interactions impact the growth and survival of competing strains, while being spatially advected by turbulent flows? By using numerical simulations of a continuum model, we study the dynamics of two antagonistic strains that are dispersed by incompressible turbulent flows in two spatial dimensions. A key parameter is the ratio of the fluid transport time to that of biological reproduction, which determines the winning organism that ultimately takes over the whole population from an initial heterogeneous state, a process known as fixation. By quantifying the probability and mean time for fixation along with the spatial structure of concentration fluctuations, we demonstrate how turbulence raises the threshold for biological nucleation and antagonism suppresses flow-induced mixing by depleting the population at interfaces. Our work highlights the unusual biological consequences of the interplay of turbulent fluid flows with antagonistic population dynamics, with potential implications for marine microbial ecology and origins of biological chirality.