Transitive reduction with vertex additions?
I don't think that any such "small" graph can exist in general. 

Let $n$ be a multiple of $4$ and pick your favorite binary string $s = \langle s_0, \dots, s_{\ell-1} \rangle$ of length $ \ell = \frac{n^2}{4}$. 

Build a directed bipartite graph $G_s = (A \cup B, E)$ on $n$ vertices where $A = \{a_0, \dots, a_{\frac{n}{2}-1}\}$ and $B = \{b_0, \dots, b_{\frac{n}{2}-1}\}$. $E$ contains the edge $(a_i,b_j) \in A \times B$ iff $s_{\frac{in}{2} + j}$ is $1$.

Let $G'_s$ be the smallest graph obtainable from $G_s$ that preserves connectivity relations.
Notice that, from $G'_s$, it is possible to recover the whole string $s$.

There are $2^\ell$ choices for $s$, which means that, for at least one graph $G_s$, the minimum number of bits needed to encode $G'_s$ must be at least $\ell = \Omega(n^2)$.

Notice that $O(m \log n)$ bits suffice to encode any graph with $m$ edges and $n$ vertices. This means that $G'_s = (V', E')$ must be such that $|E'| \cdot \log |V'| \cdot \log n = \Omega(n^2)$.