  The Kohayakawa-Nagle-R\"odl-Schacht conjecture roughly states that every
sufficiently large locally $d$-dense graph $G$ on $n$ vertices must contain at
least $(1-o(1))d^{|E(H)|}n^{|V(H)|}$ copies of a fixed graph $H$. Despite its
important connections to both quasirandomness and Ramsey theory, there are very
few examples known to satisfy the conjecture. We provide various new classes of
graphs that satisfy the conjecture. Firstly, we prove that adding an edge to a
cycle or a tree produces graphs that satisfy the conjecture. Secondly, we prove
that a class of graphs obtained by gluing complete multipartite graphs in a
tree-like way satisfies the conjecture. We also prove an analogous result with
odd cycles replacing complete multipartite graphs.
