In a reinforced Galton-Watson process with reproduction law $\boldsymbol{\nu}$ and memory parameter $q\in(0,1)$, the number of children of a typical individual either, with probability $q$, repeats that of one of its forebears picked uniformly at random, or, with complementary probability $1-q$, is given by an independent sample from $\boldsymbol{\nu}$. We estimate the average size of the population at a large generation, and in particular, we determine explicitly the Malthusian growth rate in terms of $\boldsymbol{\nu}$ and $q$. Our approach via the analysis of transport equations owns much to works by Flajolet and co-authors.