Let A n = ε_n • • • ε_1 , where (ε n) n≥1 is a sequence of independent random matrices taking values in GL d (R), d ≥ 2, with common distribution µ. In this paper, under standard assumptions on µ (strong irreducibility and proximality), we prove Berry-Esseen type theorems for log(||A n||) when µ has a polynomial moment. More precisely, we get the rate √ log n/ √ n when µ has a moment of order 3 and the rate 1/ √ n when µ has a moment of order 4, which significantly improves earlier results in this setting.