Polynomial-in-$r$ bounds for forbidden traces of uniform hypergraphs

We give a general principle that converts fixed-uniformity bounds for forbidden traces into bounds with polynomial dependence on the uniformity. More precisely, let $H$ be a fixed set system on $h\ge 1$ vertices, and suppose that, for some $\alpha \ge 0$, ${\mathrm{ex}}_j(m,\mathrm{Tr}(H))=O_{H,j}(m^{\alpha })$ for every fixed integer $j\ge 1$. Then, for every $\epsilon >0$, there is a constant $C_{H,\alpha ,\epsilon }$ such that $${\mathrm{ex}}_r(m,\mathrm{Tr}(H))\le C_{H,\alpha ,\epsilon }{r}^{h-1-\alpha +\epsilon }{m}^{\alpha }$$ for all $m\ge r\ge 2$. In particular, for trace-$C_4$-free hypergraphs and every $\epsilon >0$ there is a constant $C_{\epsilon }$ such that $${\mathrm{ex}}_r(m,\mathrm{Tr}(C_4))\le C_{\epsilon }{r}^{3\mathrm{/}2+\epsilon }{m}^{3\mathrm{/}2}$$ for all $m\ge r\ge 2$. We also construct trace-$C_4$-free $r$-graphs showing that $${\mathrm{ex}}_r(m,\mathrm{Tr}(C_4))\ge c{r}^{1\mathrm{/}2}{m}^{3\mathrm{/}2}$$ for an absolute constant $c>0$, for every fixed $r\ge 3$ and all sufficiently large $m$ (depending on $r$).