Supersonic turbulence results in strong density fluctuations in the interstellar medium (ISM), which have a profound effect on the chemical structure. Particularly useful probes of the diffuse ISM are the ArH+, OH+, H2O+ molecular ions, which are highly sensitive to fluctuations in the density and the H2 abundance. We use isothermal magnetohydrodynamic simulations of various sonic Mach numbers, Ms, and density decorrelation scales, ydec, to model the turbulent density field. We post process the simulations with chemical models and obtain the probability density functions (PDFs) for the H2, ArH+, OH+, and H2O+ abundances. We find that the PDF dispersions increases with increasing Ms and ydec, as the magnitude of the density fluctuations increases, and as they become more coherent. Turbulence also affects the median abundances: when Ms and ydec are high, low-density regions with low H2 abundance become prevalent, resulting in an enhancement of ArH+ compared to OH+ and H2O+. We compare our models with Herschel observations. The large scatter in the observed abundances, as well as the high observed ArH+/OH+ and ArH+/H2O+ ratios are naturally reproduced by our supersonic (Ms = 4.5), large decorrelation scale (ydec=0.8) model, supporting a scenario of a largescale turbulence driving. The abundances also depend on the ultraviolet intensity, cosmic-ray ionization rate, and the cloud column density, and the observed scatter may be influenced by fluctuations in these parameters.