  We prove the equidistribution of the Hodge locus for certain non-isotrivial,
polarized variations of Hodge structure of weight $2$ with $h^{2,0}=1$ over
complex, quasi-projective curves. Given some norm condition, we also give an
asymptotic on the growth of the Hodge locus. In particular, this implies the
equidistribution of elliptic fibrations in quasi-polarized, non-isotrivial
families of $K3$ surfaces.
