  Let the bipartite Tur\'an number $ex(m,n,H)$ of a graph $H$ be the maximum
number of edges in an $H$-free bipartite graph with two parts of sizes $m$ and
$n$, respectively. In this paper, we prove that $ex(m,n,C_{2t})=(t-1)n+m-t+1$
for any positive integers $m,n,t$ with $n\geq m\geq t\geq \frac{m}{2}+1$. This
confirms the rest of a conjecture of Gy\"{o}ri \cite{G97} (in a stronger form),
and improves the upper bound of $ex(m,n,C_{2t})$ obtained by Jiang and Ma
\cite{JM18} for this range. We also prove a tight edge condition for
consecutive even cycles in bipartite graphs, which settles a conjecture in
\cite{A09}. As a main tool, for a longest cycle $C$ in a bipartite graph, we
obtain an estimate on the upper bound of the number of edges which are incident
to at most one vertex in $C$. Our two results generalize or sharpen a classical
theorem due to Jackson \cite{J85} in different ways.
