We study the shape of the gas-phase mass-metallicity relation (MZR) of a combined sample of present-day dwarf and high-mass star-forming galaxies using IZI, a Bayesian formalism for measuring chemical abundances presented in Blanc et al. 2015. We observe a characteristic stellar mass scale at $M_* simeq 10^{9.5}$M$_{odot}$, above which the ISM undergoes a sharp increase in its level of chemical enrichment. In the $10^{6}-10^{9.5}$M$_{odot}$ range the MZR follows a shallow power-law ($Zpropto M^{alpha}_*$) with slope $alpha=0.14pm0.08$. At approaching $M_* simeq 10^{9.5}$M$_{odot}$ the MZR steepens significantly, showing a slope of $alpha=0.37pm0.08$ in the $10^{9.5}-10^{10.5}$M$_{odot}$ range, and a flattening towards a constant metallicity at higher stellar masses. This behavior is qualitatively different from results in the literature that show a single power-law MZR towards the low mass end. We thoroughly explore systematic uncertainties in our measurement, and show that the shape of the MZR is not induced by sample selection, aperture effects, a changing N/O abundance, the adopted methodology used to construct the MZR, secondary dependencies on star formation activity, nor diffuse ionized gas (DIG) contamination, but rather on differences in the method used to measure abundances. High resolution hydrodynamical simulations can qualitatively reproduce our result, and suggest a transition in the ability of galaxies to retain their metals for stellar masses above this threshold. The MZR characteristic mass scale also coincides with a transition in the scale height and clumpiness of cold gas disks, and a typical gas fraction below which the efficiency of star formation feedback for driving outflows is expected to decrease sharply.
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