On non-degenerate Turán problems for expansions

The $r$-uniform expansion $F^{(r)+}$ of a graph $F$ is obtained by enlarging each edge with $r-2$ new vertices such that altogether we use $(r-2)\mathrm{\mid }E(F)\mathrm{\mid }$ new vertices. Two simple lower bounds on the largest number ${\mathrm{e}\mathrm{x}}_r(n,F^{(r)+})$ of $r$-edges in $F^{(r)+}$-free $r$-graphs are $\mathrm{\Omega }(n^{r-1})$ (in the case $F$ is not a star) and $\mathrm{e}\mathrm{x}(n,K_r,F)$, which is the largest number of $r$-cliques in $n$-vertex $F$-free graphs. We prove that ${\mathrm{e}\mathrm{x}}_r(n,F^{(r)+})=\mathrm{e}\mathrm{x}(n,K_r,F)+O(n^{r-1})$. The proof comes with a structure theorem that we use to determine ${\text{ex}}_r(n,F^{(r)+})$ exactly for some graphs $F$, every $r\chi (F)$ and sufficiently large $n$.