  Advancing the sparse regularity method, we prove one-sided and two-sided
regularity inheritance lemmas for subgraphs of bijumbled graphs, improving on
results of Conlon, Fox and Zhao [Adv. Math. 256 (2014), 206--290]. These
inheritance lemmas also imply improved $H$-counting lemmas for subgraphs of
bijumbled graphs, for some $H$.
