  We prove estimates for $\mathbb{E} \| X: \ell_{p'}^n \to \ell_q^m\|$ for
$p,q\ge 2$ and any random matrix $X$ having the entries of the form
$a_{ij}Y_{ij}$, where $Y=(Y_{ij})_{1\le i\le m, 1\le j\le n}$ has i.i.d.
isotropic log-concave rows. This generalises the result of Gu\'edon, Hinrichs,
Litvak, and Prochno for Gaussian matrices with independent entries. Our
estimate is optimal up to logarithmic factors. As a byproduct we provide the
analogue bound for $m\times n$ random matrices, which entries form an
unconditional vector in $\mathbb{R}^{mn}$. We also prove bounds for norms of
matrices which entries are certain Gaussian mixtures.
