  We improve on a construction of Mestre--Shioda to produce some families of
curves $X/\mathbb{Q}$ of record rank relative to the genus $g$ of $X$. Our
first main result is that for any integer $g \geqslant 8$ with $g \equiv 2
\pmod 3$, there exist infinitely many genus $g$ hyperelliptic curves over
$\mathbb{Q}$ with at least $8g+32$ $\mathbb{Q}$-points and Mordell--Weil rank
$\geqslant 4g + 15$ over $\mathbb{Q}$. Our second main theorem is that if $g+1$
is an odd prime and $K$ contains the $g+1$-th roots of unity, then there exist
infinitely many genus $g$ hyperelliptic curves over $K$ with Mordell--Weil rank
at least $6g$ over $K$.
