  Let $ t\ge s\ge2$ be integers. Confirming a conjecture of Mader, Liu and
Montgomery [J. Lond. Math. Soc., 2017] showed that every $K_{s, t}$-free graph
with average degree $d$ contains a subdivision of a clique with at least
$\Omega(d^{\frac{s}{2(s-1)}})$ vertices. We give an improvement by showing that
such a graph contains a balanced subdivision of a clique with the same order,
where a balanced subdivision is a subdivision in which each edge is subdivided
the same number of times.
  In 1975, Erd\H{o}s asked whether the sum of the reciprocals of the cycle
lengths in a graph with infinite average degree $d$ is necessarily infinite.
Recently, Liu and Montgomery [J. Amer. Math. Soc., 2023] confirmed the
asymptotically correct lower bound on the reciprocals of the cycle lengths, and
provided a lower bound of at least $(\frac{1}{2} -o_d(1)) \log d$. In this
paper, we improve this low bound to $\left(\frac{s}{2(s-1)} -o_d(1)\right) \log
d$ for $K_{s, t}$-free graphs.
  Both proofs of our results use the graph sublinear expansion property as well
as some novel structural techniques.
