TITLE: Erdős–Stone theorem type edge density estimates for bipartite graphs?
QUESTION [5 upvotes]: The Erdős–Stone theorem theory says that the densest graph not containing a graph H (which has chromatic number r) has number of edges equal to $(r-2)/(r-1) {n \choose 2}$ asymptotically.
However, this doesn't say much for bipartite graphs (since r=2). I wanted to know what are the best results known for the densest graphs not containing a particular bipartite graph H. I'm guessing this problem is still open and hasn't been completely resolved.
This problem is easy if H is a forest, since every graph with $|E| > k|V|$ contains every forest on k vertices as a subgraph. For even cycles, I know there is a result of Bondy and Simonovits which says:
"if $|E| \geq 100k|V|^{1+1/k}$  then G contains a $C_{2l}$ for every $l$ in $[k, n^{1/k}]$."
So can someone point me to the best known results now for bipartite cyclic graphs?

REPLY [3 votes]: To add to Konrad's answer:
No construction is known for $K_{4,4}$. Neither there is a construction for $C_{2k}$ for k other than 2,3,5. There is a construction by Lazebnik and Woldar that beats the probabilistic for $C_{2k}$, though. There are some non-trivial upper bounds on bipartite graphs of bounded degree and (more generally) of bounded degeneracy. These upper bounds (and some references) can be found in the survey on dependent random choice by Fox and Sudakov.