We give an improved algorithm for counting the number of $1324$-avoiding permutations, resulting in 5 further terms of the generating function. We analyse the known coefficients and find compelling evidence that unlike other classical length-4 pattern-avoiding permutations, the generating function in this case does not have an algebraic singularity. Rather, the number of 1324-avoiding permutations of length $n$ behaves as $$B\cdot μ^n \cdot μ_1^{n^σ} \cdot n^g.$$ We estimate $μ=11.60 \pm 0.01,$ $σ=1/2,$ $μ_1 = 0.0398 \pm 0.0010,$ $g = -1.1 \pm 0.2$ and $B =9.5 \pm 1.0.$