  A graph is called $1$-planar if it admits a drawing in the plane such that
each edge is crossed at most once. Let $G$ be a bipartite 1-planar graph with
$n$ ($\ge 4$) vertices and $m$ edges. Karpov showed that $m\le 3n-8$ holds for
even $n\ge 8$ and $m\le 3n-9$ holds for odd $n\ge 7$. Czap, Przybylo and
\u{S}krabul\'{a}kov\'{a} proved that if the partite sets of $G$ are of sizes
$x$ and $y$, then $m\le 2n+6x-12$ holds for $2\leq x\leq y$, and conjectured
that $m\le 2n+4x-12$ holds for $x\ge 3$ and $y\ge 6x-12$. In this paper, we
settle their conjecture and our result is even under a weaker condition $2\le
x\le y$.
