  We expand Conlon's random algebraic construction to show that for any odd
number $k \geq 3$ exists a natural number $c_k$ (the same as Conlon's) such
that $\operatorname{ex}(n^a,n,\theta_{k,c_k}) = \Omega_{k,a}((n^{1 +
a})^{\frac{k + 1}{2k}})$, with $a \in [\frac{k - 1}{k + 1}, 1)$. Where given a
graph $H$, we denote by $\operatorname{ex}(n,m,H)$ the maximum number of edges
an $H-$free bipartite graph can have when the cardinalities of its parts are
$n$ and $m$. Also, we denote with $\theta_{k,l}$ the graph where two vertices
are connected through $l$ disjoint paths of length $k$.
