the largest graph with 7 vertices and girth 4 
This is my graph. Am I right? And if there is any trick for such kind of problem? Thank you!
 A: How about adding a couple more edges:

Basically if there are two vertices which do not share any neighbours, you cannot reduce the girth below $4$ by joining them.
In this case the degree-2 vertices stood out as underused.

As observed by Misha Lavrov in comments, this is actually the complete bipartite graph $K_{4,3}$:

It seems likely that the most edges for a girth-$4$ graph of a given number of vertices $v$ is always the appropriate complete bipartite graph with parts as nearly matched as possible, i.e. $K_{n,n}$ or $K_{n{+}1,n}$
A: 
Proposition: If a cycle has a length of at least $2(n-1)$, where $n$ is the girth, you can divide it into two cycles (one of which has a length equal to the girth) by adding an edge connecting two nodes in the cycle that are not in the same subcycle.

This works because you create a cycle with length equal to the girth and because you don't shorten any cycle with a length equal to the girth.
In your case, you can add an edge between the top right and the bottom left and the top left in the bottom right because you have a cycle of six in the left six points.
