  We show that plane bipolar posets (i.e., plane bipolar orientations with no
transitive edge) and transversal structures can be set in correspondence to
certain (weighted) models of quadrant walks, via suitable specializations of a
bijection due to Kenyon, Miller, Sheffield and Wilson. We then derive exact and
asymptotic counting results. In particular we prove (computationally and then
bijectively) that the number of plane bipolar posets on $n+2$ vertices equals
the number of plane permutations of size $n$. Regarding transversal structures,
for each $v\geq 0$ we consider $t_n(v)$ the number of such structures with
$n+4$ vertices and weight $v$ per quadrangular inner face (the case $v=0$
corresponds to having only triangular inner faces). We obtain a recurrence to
compute $t_n(v)$, and an asymptotic formula that for $v=0$ gives $t_n(0)\sim c\
\!(27/2)^nn^{-1-\pi/\mathrm{arccos}(7/8)}$ for some $c>0$, which also ensures
that the associated generating function is not D-finite.
