  We construct an infinite family $\{ C_{n,k}\}_{k=1}^{\infty}$ of corks of
Mazur type satisfying $2n\leq \mathrm{sc}^{\mathrm{sp}}(C_{n,k})\leq
O(n^{3/2})$ for any positive integer $n$. Furthermore, using these corks, we
construct an infinite family $\{(W_{n,k},W'_{n,k})\}_{k=1}^{\infty}$ of exotic
pairs of $4$-manifolds with boundary whose special shadow-complexities satisfy
the above inequalities. We also discuss exotic pairs with small
shadow-complexity.
