Turán numbers of triangular pyramids

Let ${\text{TP}}_k$ denote the triangular pyramid with $k$ layers. For a graph family $\mathcal{F}$, let $\text{ex}(n,\mathcal{F})$ be the maximum number of edges in an $n$-vertex graph containing no member of $\mathcal{F}$ as a subgraph. For a graph $H$, let $\text{cJ}(H)$ be the family of subgraph-minimal graphs among all such graphs $H-I$, where $I\subseteq V(H)$ is an independent set. For every integer $k\ge 4$ and all sufficiently large $n$, we prove that $$\text{ex}(n,{\text{TP}}_k)=\underset{0\le m\le n}{\max }\{m(n-m)+\text{ex}(m,\text{cJ}({\text{TP}}_k))\}+O_k(n)=\lfloor n^2\mathrm{/}4\rfloor +{\mathrm{\Theta }}_k(n^{4\mathrm{/}3})\mathrm{.}$$ Moreover, for $k=4$, we characterize all extremal graphs and show that $$\text{ex}(n,{\text{TP}}_4)=\underset{0\le m\le n}{\max }\{m(n-m)+\text{ex}(m,\text{cJ}({\text{TP}}_4)\left)\right\}$$ for sufficiently large $n$.