$(2019/3)^2$

The construction is as follows: take the case where each island has $673$ towns, and on each island label the towns as $0, 1, \dotsc, 672$. Let islands be $A, B$ and $C$. For towns with labels $a$, $b$ and $c$ that are in island $A$, $B$ and $C$ respectively, we join them with a flight route if and only if $c\equiv a-b\enspace(\text{mod}\enspace673)$. This way, each pair of towns is connected by a unique flight route since the if we know any two of $a, b$ or $c$, we can use the congruence relation to solve for the remaining one. (Surprisingly) Construction is the only hard part of the problem. For any two islands say $A$ and $B$, the total number of flight routes is at most $|A||B|$ since each pair of towns is connected by at most one flight route. Since minimum of $|A||B|, |B||C|$ and $|C||A|$ is maximized when $|A|=|B|=|C|=673$, the desired bound follows.