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This paper reports the first measurement of the relationship between turbulent velocity and cloud size in the diffuse circumgalactic medium (CGM) in typical galaxy halos at redshift z ≈ 0.4–1. Through spectrally resolved absorption profiles of a suite of ionic transitions paired with careful ionization analyses of individual components, cool clumps of size as small as l _cl ∼ 1 pc and density lower than n _H = 10 ^−3 cm ^−3 are identified in galaxy halos. In addition, comparing the line widths between different elements for kinematically matched components provides robust empirical constraints on the thermal temperature T and the nonthermal motions b _NT , independent of the ionization models. On average, b _NT is found to increase with l _cl following ${b}_{\mathrm{NT}}\propto {l}_{\mathrm{cl}}^{0.3}$ over three decades in spatial scale from l _cl ≈ 1 pc to l _cl ≈ 1 kpc. Attributing the observed b _NT to turbulent motions internal to the clumps, the best-fit b _NT – l _cl relation shows that the turbulence is consistent with Kolmogorov at <1 kpc with a roughly constant energy transfer rate per unit mass of ϵ ≈ 0.003 cm ^2 s ^−3 and a dissipation timescale of ≲100 Myr. No significant difference is found between massive quiescent and star-forming halos in the sample on scales less than 1 kpc. While the inferred ϵ is comparable to what is found in C iv absorbers at high redshift, it is considerably smaller than observed in star-forming gas or in extended line-emitting nebulae around distant quasars. A brief discussion of possible sources to drive the observed turbulence in the cool CGM is presented.