Extremal related question regarding Intersection of neighbourhood of vertices
This problem has been solved in the undirected case by Füredi, New asymptotics for bipartite Turán numbers, who showed that for constant $k$, the optimal bound is
$$
\frac{1}{2} \sqrt{k} n^{3/2} + O(n^{4/3}),
$$
when counting undirected edges. When counting directed edges, you have to double the number. The lower bound also holds in the directed case &mdash; it is given by an explicit construction based on finite fields.

Following Füredi's footsteps, we can generalize the upper bound to the directed case. Denote by $P_2$ a path consisting of two directed edges pointing at the central cases. The number of copies of $P_2$ in your graph is at most $k \binom{n}{2}$, since for every choice of the outer vertices there are at most $k$ choices of the inner vertex. On the other hand, the number of copies of $P_2$ is also $\sum_v \binom{d_{in}(v)}{2}$, where $d_{in}$ is the in-degree of $v$, since for every $v$, we can choose any two vertices in its incoming neighborhood to create a copy of $P_2$. Convexity now shows that $\sum_v \binom{d_{in}(v)}{2}$ is minimized when all in-degrees are the same. If the graph contains $m$ edges, then the total in-degree is $m$, and so
$$
k \binom{n}{2} \geq n \binom{m/n}{2},
$$
from which we can deduce $m &lt; \sqrt{k} n^{3/2} + n/2$.