A bipartite graph $(A,B)$ is $(p, \beta)$-jumbled if for all subsets $A'\subseteq A$ and $B'\subseteq B$ we have that $\left|\mathrm{E}(A',B')-p|A'||B'|\right|\leq \beta \sqrt{|A'||B'|}$. A easy corollary of Szemerédi regularity lemma is that a dense graph (with $\Omega(n^2)$ edges) contains a $\Omega(n)$-sized balanced bipartite graph which is $(p,o(n))$-jumbled, for some $p>0$. In fact, any dense graph can essentially be partitioned into such jumbled bipartite graphs.

My question is the following: Do dense graphs contain jumbled balanced linear sized bipartite graphs with smaller values of $\beta$? Say $\beta=O(n^{1-\varepsilon})$?

In case this is impossible, what about polynomially sized jumbled graphs? That is, let $m=\Omega(n^{\varepsilon})$. Does a dense graph necessarily contain a balanced bipartite graph on $m$ vertices which is $(p,m^{1-\gamma})$-jumbled, for some $p>0$?