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We report observational evidence of highly turbulent ionized gas kinematics in a sample of 20 Lyman continuum (LyC) emitters (LCEs) at low redshift ($z\sim 0.3$). Detailed Gaussian modeling of optical emission line profiles in high-dispersion spectra consistently shows that both bright recombination and collisionally excited lines can be fitted as one or two narrow components with intrinsic velocity dispersion of $\sigma$ $\sim$ 40-100 km s$^{-1}$, in addition to a broader component with $\sigma \sim$ 100-300 km s$^{-1}$, which contributes up to $\sim$40% of the total flux and is preferentially blueshifted from the systemic velocity. We interpret the narrow emission as highly ionized gas close to the young massive star clusters and the broader emission as a signpost of unresolved ionized outflows, resulting from massive stars and supernova feedback. We find a significant correlation between the width of the broad emission and the LyC escape fraction, with strong LCEs exhibiting more complex and broader line profiles than galaxies with weaker or undetected LyC emission. We provide new observational evidence supporting predictions from models and simulations; our findings suggest that gas turbulence and outflows resulting from strong radiative and mechanical feedback play a key role in clearing channels through which LyC photons escape from galaxies. We propose that the detection of blueshifted broad emission in the nebular lines of compact extreme emission-line galaxies can provide a new indirect diagnostic of Lyman photon escape, which could be useful to identify potential LyC leakers in the epoch of reionization with the JWST.