We study how the degrees of irrationality of moduli spaces of polarized K3 surfaces grow with respect to the genus
g
g
. We prove that the growth is bounded by a polynomial function of degree
14
+
ε
14+\varepsilon
for any
ε
>
0
\varepsilon >0
and, for three sets of infinitely many genera, the bounds can be refined to polynomials of degree
10
10
. The main ingredients in our proof are the modularity of the generating series of Heegner divisors due to Borcherds and its generalization to higher codimensions due to Kudla, Millson, Zhang, Bruinier, and Westerholt-Raum. For special genera, the proof is also built upon the existence of K3 surfaces associated Hodge theoretically with certain cubic fourfolds, Gushel–Mukai fourfolds, and hyperkähler fourfolds.