Let $ex(n, H)$ denote the maximum number of edges of a graph on $n$ vertices not containing a copy of $H$. The exact bounds are difficult already if you forbid complete bipartite graphs $K_{m,n}$.
Erdös, Rényi, Sós (1954) showed that $$ex(n,K_{2,2}) \sim \frac{1}{2}n^{3/2}.$$
According to the classical Kövári-Sós-Turán Theorem (1954), $$ex(n,K_{t,s}) \leq c_{s,t} n^{2-\frac{1}{t}}$$ for $s\geq t\geq 2$,
while a random construction gives the lower bound $$ex(n,K_{t,s})\geq cn^{2-\frac{s+t-2}{st-1}}.$$
Brown (1966) showed Kövári-Sós-Turán is tight for $s=t=3$: $$ex(n,K_{3,3}) \geq cn^{2-\frac{1}{3}},$$ and Füredi (1996) proved that the constant in Brown's construction is optimal, giving $$ex(n,K_{3,3}) \sim \frac{1}{2}n^{2-\frac{1}{3}}.$$
Alon, Kollár, Rónyai, Szabó (1995, 1999) showed that for each $t\geq 2$ there exists $c_t>0$ such that for all $s\geq(t-1)!+1$,
$$ex(n,K_{t,s}) \geq c_tn^{2-\frac{1}{t}},$$ thus matching Kövári-Sós-Turán asymptotically.
I'm not sure if a construction matching Kövári-Sós-Turán has been found for the case $K_{4,4}$. There must also be more known than what is mentioned here. But as you can see, somewhat more than a just a little more is known, although the gaps in our knowledge are still huge here.
The appropriate search term is "supersaturation". I think Theorem 11.1 in
Furedi, Z., Simonovits, M. (2013). The history of degenerate (bipartite) extremal graph problems. In Erdos Centennial (pp. 169-264). arxiv:1306.5167
implies the existence of a constants $c_1=c_1(r,s)$ and $c_2=c_2(r,s)$ such that a graph with $m>c_2n^{2-1/r}$ edges contains at least
$c_1c_2^{rs}n^r$
copies of $K_{r,s}$.
Best Answer
Proving the conjecture for host graphs $G$ with $\Delta/\delta$ bounded is equivalent to proving the full conjecture. One may even assume that the host graph $G$ is both vertex- and edge-transitive, a result of Szegedy that can be found at https://arxiv.org/abs/1504.00858.