  A transversal in an $n \times n$ latin square is a collection of $n$ entries
not repeating any row, column, or symbol. Kwan showed that almost every $n
\times n$ latin square has $\bigl((1 + o(1)) n / e^2\bigr)^n$ transversals as
$n \to \infty$. Using a loose variant of the circle method we sharpen this to
$(e^{-1/2} + o(1)) n!^2 / n^n$. Our method works for all latin squares
satisfying a certain quasirandomness condition, which includes both random
latin squares with high probability as well as multiplication tables of
quasirandom groups.
