One of the fundamental problems in extremal graph theory is the Turan problem which asks for the largest number of edges in a simple $n$-vertex graph $G$ not containing a given graph $H$ as a subgraph. This largest number is called the Turan number of $H$ and is denoted by $ex(n,H)$. The Turan number is well understood for nonbipartite graphs. For general bipartite graphs, however, the problem remains very difficult.
After a brief introduction to the current status of the Turan problem for bipartite graphs, we discuss some Turan type results for bipartite graphs obtained through edge subdivision. One of these is a so-called topological clique: a graph obtained from subdividing edges of a complete graph. For each $t$ and sufficiently large $n$, we show that every $n$-vertex graph with at least $a(t) n^{1+\epsilon}$ edges, where $a(t)$ is a constant depending on $t$, contains a subdivision of $K_t$ in which each edge of $K_t$ is subdivided at most $c\log (1/\epsilon)/\epsilon$ times, where $c$ is an absolute constant. This improves an earlier result of Kostochka and Pyber. We also show that if H is obtained from iteratively adding a path of odd length at least $2k-1$ between a pair of adjacent vertices then $ex(n,H)=O(n^{1+1/k})$, which substantially generalizes the known result on the Turan numbers of even cycles..

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