  By applying Chinese remainder theorem for coprime polynomials and the
"creative microscoping" method recently introduced by the author and Zudilin,
we establish parametric generalizations of three $q$-supercongruences modulo
the fourth power of a cyclotomic polynomial. The original $q$-supercongruences
then follow from these parametric generalizations by taking the limits as the
parameter tends to $1$ (l'H\^opital's rule is utilized here). In particular, we
prove a complete $q$-analogue of the (J.2) supercongruence of Van Hamme and a
complete $q$-analogue of a "divergent" Ramanujan-type supercongruence, thus
confirming two recent conjectures of the author. We also put forward some
related conjectures, including a $q$-supercongruence modulo the fifth power of
a cyclotomic polynomial.
