  In 1997, Van Hamme conjectured 13 Ramanujan-type supercongruences. All of the
13 supercongruences have been confirmed by using a wide range of methods. In
2015, Swisher conjectured Dwork-type supercongruences related to the first 12
supercongruences of Van Hamme. Here we prove that the (C.3) and (J.3)
supercongruences of Swisher are true modulo $p^{3r}$ (the original modulus is
$p^{4r}$) by establishing $q$-analogues of them. Our proof will use the
creative microscoping method, recently introduced by the author in
collaboration with Zudilin. We also raise conjectures on $q$-analogues of an
equivalent form of the (M.2) supercongruence of Van Hamme, partially answering
a question at the end of [Adv. Math. 346 (2019), 329--358].
