  Let $A_n= \varepsilon_n \cdots \varepsilon_1$, where $(\varepsilon_n)_{n \geq
1}$ is a sequence of independent random matrices taking values in $
GL_d(\mathbb R)$, $d \geq 2$, with common distribution $\mu$. In this paper,
under standard assumptions on $\mu$ (strong irreducibility and proximality), we
prove Berry-Esseen type theorems for $\log ( \Vert A_n \Vert)$ when $\mu$ has a
polynomial moment. More precisely, we get the rate $((\log n) / n)^{q/2-1}$
when $\mu$ has a moment of order $q \in ]2,3]$ and the rate $1/ \sqrt{n} $ when
$\mu$ has a moment of order $4$, which significantly improves earlier results
in this setting.
