A counterexample to sparse removal

The Turán number of a graph H, denoted ex(n,H), is the maximum number of edges in an n-vertex graph with no subgraph isomorphic to H. Solymosi [17] conjectured that if H is any graph and ex(n,H) = O(n) where α > 1, then any n-vertex graph with the property that each edge lies in exactly one copy of H has o(n) edges. This can be viewed as conjecturing a possible extension of the removal lemma to sparse graphs, and is well-known to be true when H is a non-bipartite graph, in particular when H is a triangle, due to Ruzsa and Szemerédi [16]. Using Sidon sets we exhibit infinitely many bipartite graphs H for which the conjecture is false.