Ordinary regularity gives tower upper bounds; strong regularity gives wowzer upper bounds only when the regularity sequence depends on the coarse part count

The energy-increment proof of ordinary regularity in Szemerédi regularity lemma with an equitable partition and an explicit tower-type upper bound for graphs of order at least $m_0$ iterates an exponential part-count recurrence only a bounded number of times and therefore gives a tower-type upper bound. The proof of Equitable strong regularity lemma: a very regular refinement that changes energy only slightly repeatedly invokes ordinary regularity at parameters indexed by an already enormous partition size, so when the regularity sequence genuinely depends on the coarse part count the nested iteration gives a wowzer-type upper bound. That dependence is what Induced graph removal lemma for a fixed graph does not need: the parameter it requires of its representative pairs comes from Induced counting lemma: regular edge and nonedge pairs force many induced copies and depends only on the pattern and the density threshold, so a constant sequence suffices, the nesting collapses to boundedly many applications at one fixed parameter, and the bound stays tower-type. These are upper bounds delivered by the displayed proofs, not claims of optimality.