Complete minors in graphs without sparse cuts

We show that if $G$ is a graph on $n$ vertices, with all degrees comparable to some $d=d(n)$, and without a sparse cut, for a suitably chosen notion of sparseness, then it contains a complete minor of order $$\mathrm{\Omega }\left(\sqrt{\frac{nd}{\log d}}\right)\mathrm{.}$$ As a corollary we determine the order of a largest complete minor one can guarantee in $d$-regular graphs for which the second largest eigenvalue is bounded away from $d\mathrm{/}2$, in $(d\mathrm{/}n,o(d))$-jumbled graphs, and in random $d$-regular graphs, for almost all $d=d(n)$.