We prove estimates for E}X : ℓ n p 1 Ñ ℓ m q } for p, q ě 2 and any random matrix X having the entries of the form aijYij, where Y " pYij q1ďiďm,1ďjďn has i.i.d. isotropic log-concave rows. This generalises the result of Guédon, 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 mˆn random matrices, which entries form an unconditional vector in R mn . We also prove bounds for norms of matrices which entries are certain Gaussian mixtures.