On non-degenerate Turán problems for expansions

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