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The bipartite Turán number, denoted by   ex   (   m  ,  n  ,  H   )   , is the maximum number of edges in an   H  ‐free bipartite graph with two parts of sizes   m   and   n  , respectively. In this article, we prove   ex   (   m  ,  n  ,   C   2  t)   =   (   t  −  1   )   n  +  m  −  t  +  1   for any positive integers   m  ,  n  ,  t   with   n  ≥  m  ≥  t  ≥   m  2   +  1  , which confirms (in a strong form) the unsolved case of a conjecture of Győri. Let   G  =   (   X  ,  Y  ;  E   )    be a bipartite graph with   ∣  X  ∣  =  n   and   ∣  Y  ∣  =  m  . Suppose that   n  ≥  m  ≥  2  k  +  2   for some   k  ∈  N  . We prove that   G   contains cycles of all even lengths from 4 to   2  m  −  2  k   if   e   (  G  )   ≥   (   m  −  k  −  1   )   n  +  k  +  2  . This result sharpens a theorem on long cycles in bipartite graphs due to Jackson. As a main tool, for a longest cycle   C   in a bipartite graph, we prove an upper bound of the number of edges which are incident with at most one vertex in   C  , which is a bipartite analogue of a classical theorem of Bondy. Our proof technique is structural.

Exact bipartite Turán numbers of large even cycles