  For every integer $t \ge 0$, denote by $F_5^t$ the hypergraph on vertex set
$\{1,2,\ldots, 5+t\}$ with hyperedges $\{123,124\} \cup \{34k : 5 \le k \le
5+t\}$. We determine $\mathrm{ex}(n,F_5^t)$ for every $t\ge 0$ and sufficiently
large $n$ and characterize the extremal $F_5^t$-free hypergraphs. In
particular, if $n$ satisfies certain divisibility conditions, then the extremal
$F_5^t$-free hypergraphs are exactly the balanced complete tripartite
hypergraphs with additional hyperedges inside each of the three parts
$(V_1,V_2,V_3)$ in the partition; each part $V_i$ spans a
$(|V_i|,3,2,t)$-design. This generalizes earlier work of Frankl and F\"uredi on
the Tur\'an number of $F_5:=F_5^0$.
  Our results extend a theory of Erd\H{o}s and Simonovits about the extremal
constructions for certain fixed graphs. In particular, the hypergraphs
$F_5^{6t}$, for $t\geq 1$, are the first examples of hypergraphs with
exponentially many extremal constructions and positive Tur\'an density.
