  In this paper, we study the well-known Kohayakawa-Nagle-R{\"o}dl-Schacht
(KNRS) conjecture, with a specific focus on graph subdivisions. The KNRS
conjecture asserts that for any graph $H$, locally dense graphs contain
asymptotically at least the number of copies of $H$ found in a random graph
with the same edge density. We prove the following results about
$k$-subdivisions of graphs (obtained by replacing edges with paths of length
$k+1$): (1). If $H$ satisfies the KNRS conjecture, then its
$(2k-1)$-subdivision satisfies Sidorenko's conjecture, extending a prior result
of Conlon, Kim, Lee and Lee; (2). If $H$ satisfies the KNRS conjecture, then
its $2k$-subdivision satisfies a constant-fraction version of the KNRS
conjecture; (3). If $H$ is regular and satisfies the KNRS conjecture, then its
$2k$-subdivision also satisfies the KNRS conjecture. These findings imply that
all balanced subdivisions of cliques satisfy the KNRS conjecture, improving
upon a recent result of Brada{\v c}, Sudakov and Wigerson. Our work provides
new insights into this pivotal conjecture in extremal graph theory.
