  We show that the Hausdorff dimension of $(s,t)$-Furstenberg sets is at least
$s+t/2+\epsilon$, where $\epsilon>0$ depends only on $s$ and $t$. This improves
the previously best known bound for $2s<t\le 1+\epsilon(s,t)$, in particular
providing the first improvement since 1999 to the dimension of classical
$s$-Furstenberg sets for $s<1/2$. We deduce this from a corresponding
discretized incidence bound under minimal non-concentration assumptions, that
simultaneously extends Bourgain's discretized projection and sum-product
theorems. The proofs are based on a recent discretized incidence bound of
T.~Orponen and the first author and a certain duality between $(s,t)$ and
$(t/2,s+t/2)$-Furstenberg sets.
