Unbalanced Zarankiewicz problem for bipartite subdivisions with applications to incidence geometry

For a bipartite graph $H$, its linear threshold is the smallest real number $\sigma$ such that every bipartite graph $G=(U\bigsqcup V,E)$ with unbalanced parts $\mathrm{\mid }V\mathrm{\mid }\gtrsim \mathrm{\mid }U{\mathrm{\mid }}^{\sigma }$ and without a copy of $H$ must have a linear number of edges $\mathrm{\mid }E\mathrm{\mid }\lesssim \mathrm{\mid }V\mathrm{\mid }$. We prove that the linear threshold of the complete bipartite subdivision graph $K_{s,t}^{\mathrm{\prime }}$ is at most ${\sigma }_s=2-1\mathrm{/}s$. Moreover, we show that any $\sigma <{\sigma }_s$ is less than the linear threshold of $K_{s,t}^{\mathrm{\prime }}$ for sufficiently large $t$ (depending on $s$ and $\sigma$). Some geometric applications of this result are given: we show that any $n$ points and $n$ lines in the complex plane without an $s$-by-$s$ grid determine $O(n^{4\mathrm{/}3-c})$ incidences for some constant $c>0$ depending on $s$; and for certain pairs $(p,q)$, we establish nontrivial lower bounds on the number of distinct distances determined by $n$ points in the plane under the condition that every $p$ points determine at least $q$ distinct distances.