Cycles in dense digraphs

Let G be a digraph (without parallel edges) such that every directed cycle has length at least four; let β(G) denote the size of the smallest subset X ⊆ E(G) such that G \ X has no directed cycles, and let γ(G) be the number of unordered pairs {u, v} of vertices such that u, v are nonadjacent in G. It is easy to see that if γ(G) = 0 then β(G) = 0; what can we say about β(G) if γ(G) is bounded? We prove that in general β(G) ≤ γ(G). We conjecture that in fact β(G) ≤ 12γ(G) (this would be best possible if true), and prove this conjecture in two special cases: • when V (G) is the union of two cliques, • when the vertices of G can be arranged in a circle such that if distinct u, v, w are in clockwise order and uw is a (directed) edge, then so are both uv, vw.