  We prove an effective estimate with a power saving error term for the number
of square-tiled surfaces in a connected component of a stratum of quadratic
differentials whose vertical and horizontal foliations belong to prescribed
mapping class group orbits and which have at most $L$ squares. This result
strengthens asymptotic counting formulas in work of Delecroix, Goujard, Zograf,
Zorich, and the author.
