  This paper is concerned with the existence of normalized solutions of the
nonlinear Schr\"odinger equation \[
  -\Delta u+V(x)u+\lambda u = |u|^{p-2}u \qquad\text{in $\mathbb{R}^N$} \] in
the mass supercritical and Sobolev subcritical case $2+\frac{4}{N}<p<2^*$. We
prove the existence of a solution $(u,\lambda)\in
H^1(\mathbb{R}^N)\times\mathbb{R}^+$ with prescribed $L^2$-norm $\|u\|_2=\rho$
under various conditions on the potential $V:\mathbb{R}^N\to\mathbb{R}$,
positive and vanishing at infinity, including potentials with singularities.
The proof is based on a new min-max argument.
