The ionized gas kinematics of low-mass starburst galaxies is a tracer of galaxy interactions and feedback processes, which are key for understanding massive star formation, chemical enrichment, and galaxy evolution. We study the ionized gas kinematics and outflow properties in a sample of Lyman-Break Analogs (LBAs) at z~0.1-0.3, characterized by their compact morphologies, high UV luminosities, and strong emission lines, which are common at higher redshifts. We use high-resolution VLT/X-Shooter spectra of 14 compact, UV-luminous LBAs to model the complex [OIII]{lambda}{lambda}4959,5007{AA} and Balmer line profiles with multi-Gaussian fits. LBAs show complex kinematics, with emission lines best reproduced by narrow ({sigma}<90km/s) and broad ({sigma}>90km/s) components in all galaxies. The narrow-line kinematics is highly turbulent, likely driven by massive star-forming clumps. The luminosities and line ratios of the narrow components are typical of giant HII regions. We interpret the broader components as ionized outflows driven by strong winds of massive stars and supernovae. In galaxies with highly complex profiles and disturbed morpholo- gies, ongoing interactions or mergers are found to contribute to the broad components. We find outflow velocities in the range vout~200km/s to 500km/s. Simple models yield outflow mass rates of 0.20-2.72M_{sun}_/yr and mass-loading factors {eta}~0.03-0.81. We find that {eta} shows a mild increasing trend at lower stellar masses, in agreement with previous observational studies and predictions from FIRE-2 and Illustris-TNG simulations. Compact starburst morphologies may modulate the {eta}-M* relation, showing a strong {SIGMA}_SFR_-{eta} correlation, i.e., more compact starbursts drive stronger outflows. We find a good agreement with similar findings in star-forming galaxies at high-redshift (z~2-9), including those from recent JWST observations. Our results highlight the relevance of detailed studies of the ionized gas kinematics in local UV-compact starbursts to improve our understanding of feedback processes in low-mass, rapidly star-forming galaxies