  Given integers $r > t \ge 1$ and a real number $p > 0$, the $(t,p)$-norm
$\left\lVert \mathcal{H} \right\rVert_{t,p}$ of an $r$-graph $\mathcal{H}$ is
the sum of the $p$-th power of the degrees $d_{\mathcal{H}}(T)$ over all
$t$-subsets $T \subset V(\mathcal{H})$. We conduct a systematic study of the
Tur\'{a}n-type problem of determining $\mathrm{ex}_{t,p}(n,\mathcal{F})$, which
is the maximum of $\left\lVert \mathcal{H} \right\rVert_{t,p}$ over all
$n$-vertex $\mathcal{F}$-free $r$-graphs $\mathcal{H}$.
  We establish several basic properties for the $(t,p)$-norm of $r$-graphs,
enabling us to derive general theorems from the recently established framework
in~\cite{CL24} that are useful for determining
$\mathrm{ex}_{t,p}(n,\mathcal{F})$ and proving the corresponding stability.
  We determine the asymptotic value of $\mathrm{ex}_{t,p}(n,H_{F}^{r})$ for all
feasible combinations of $(r,t,p)$ and for every graph $F$ with chromatic
number greater than $r$, where $H_{F}^{r}$ represents the expansion of $F$. In
the case where $F$ is edge-critical and $p \ge 1$, we establish strong
stability and determine the exact value of $\mathrm{ex}_{t,p}(n,H_{F}^{r})$ for
all sufficiently large $n$. These results extend the seminal theorems of
Erd\H{o}s--Stone--Simonovits, Andr\'{a}sfai--Erd\H{o}s--S\'{o}s,
Erd\H{o}s--Simonovits, and a classical theorem of Mubayi.
  For the $3$-uniform generalized triangle $F_5$, we determine the exact value
of $\mathrm{ex}_{2,p}(n,F_5)$ for all $p \ge 1$ and its asymptotic value for
all $p \in [1/2, 1]\cup \{k^{-1} \colon k \in 6\mathbb{N}^{+}+\{0,2\}\}$. This
extends old theorems of Bollob\'{a}s, Frankl--F\"{u}redi, and a recent result
of Balogh--Clemen--Lidick\'{y}. Our proofs utilize results on the graph
inducibility problem, Steiner triple systems, and the feasible region problem
introduced by Liu--Mubayi.
