  Using the $q$-Wilf--Zeilberger method and a $q$-analogue of a "divergent"
Ramanujan-type supercongruence, we give several $q$-supercongruences modulo the
fourth power of a cyclotomic polynomial. One of them is a $q$-analogue of a
supercongruence recently proved by Wang: for any prime $p>3$, $$
\sum_{k=0}^{p-1} (3k-1)\frac{(\frac{1}{2})_k (-\frac{1}{2})_k^2
}{k!^3}4^k\equiv p-2p^3 \pmod{p^4}, $$ where $(a)_k=a(a+1)\cdots (a+k-1)$ is
the Pochhammer symbol.
