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Application of in-plane uniaxial stress to the quasi two-dimensional strongly correlated system Sr₂RuO₄ results in substantial changes of the physical properties, including a more than doubled superconducting transition temperature as well as normal state transport properties deviating from conventional Fermi-liquid behavior. Both observations are attributed to a nearby Lifshitz transition, associated with moving the Fermi energy through a van Hove singularity (vHs). ¹⁷O NMR measurements reveal that the Lifshitz point is accompanied by strong field and temperature dependence of the spin susceptibility, persisting to temperatures less than 100 mK. Our NMR measurements indicate that the non-Fermi-liquid behavior is strongly influenced by the proximity of the Lifshitz transition and a large Stoner enhancement, in a system where three energy scales are nearly balanced: width of vHs, as well as Zeeman and thermal energies.