Before revision:  This is essentially best possible because  families of pairwise intersecting arrangements of n pseudocircles with p_3/n^2 \to 2/ 3 as n \infty are known.
After revision:  This is essentially best possible because there are families of pairwise intersecting arrangements of n pseudocircles with p_3/n^2 \to 2/ 3 as n \infty are known.