  In this paper, we obtain new lower and upper bounds for the problem of
bipartite biregular cages. Moreover, for girth $6$, we give the exact
parameters of the $(m,n;6)$-bipartite biregular cages when $n\equiv -1$ $\pmod
m$ using the existence of Steiner System system $S(2,k=m,v=1+n(m-1)+m)$.
  For girth $g=2r$ and $r=\{4,6,8\}$, we use results on $t$-good structures
given by ovoids, spreads and sub-polygons in generalized polygons to obtain
$(m,n;2r)$-bipartite biregular graphs. We emphasize that, as we improve the
lower bounds on the order of these graphs, we also prove that some of them are
$(m,n;2r)$-bipartite biregular cages.
  In particular, we construct relatively small bipartite biregular graphs from
a special class of generalized quadrangles and hexagons. In a special case, we
show that the graph obtained is actually a $(3,4;8)$-bipartite biregular cage
on $56$ vertices.
