  In this paper, a new $q$-supercongruence with two free parameters modulo the
fourth power of a cyclotomic polynomial is obtained. Our main auxiliary tools
are Watson's $_8\phi_7$ transformation formula for basic hypergeometric series,
the `creative microscoping' method recently introduced by Guo and Zudilin and
the Chinese remainder theorem for coprime polynomials. By taking suitable
parameter substitutions in the established $q$-supercongruence, some nice
congruences involving the Bernoulli numbers are derived.
