We combine supersymmetric localization with the numerical conformal bootstrap to bound the scaling dimension and OPE coefficient of the lowest-dimension unprotected operator in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SU(N) super-Yang-Mills theory for a wide range of N and Yang-Mills couplings gYM. We find that our bounds are approximately saturated by weak coupling results at small gYM. Furthermore, at large N our bounds interpolate between integrability results for the Konishi operator at small gYM and strong-coupling results, including the first few stringy corrections, for the lowest-dimension double-trace operator at large gYM. In particular, our scaling dimension bounds describe the level splitting between the single- and double-trace operators at intermediate coupling.