Planar point sets with forbidden $4$-point patterns and few distinct distances

We show that for any large n, there exists a set of n points in the plane with O(n²/√log n) distinct distances, such that any four points in the set determine at least five distinct distances. This answers (in the negative) a question of Erd{o}s. The proof combines an analysis by Dumitrescu of forbidden four-point patterns with an algebraic construction of Thiele and Dumitrescu (to eliminate parallelograms), as well as a randomized transformation of that construction (to eliminate most other forbidden patterns).