  We develop a sparse graph regularity method that applies to graphs with few
4-cycles, including new counting and removal lemmas for 5-cycles in such
graphs. Some applications include:
  * Every $n$-vertex graph with no 5-cycle can be made triangle-free by
deleting $o(n^{3/2})$ edges.
  * For $r \geq 3$, every $n$-vertex $r$-graph with girth greater than $5$ has
$o(n^{3/2})$ edges.
  * Every subset of $[n]$ without a nontrivial solution to the equation $x_1 +
x_2 + 2x_3 = x_4 + 3x_5$ has size $o(\sqrt{n})$.
