  Let $\Phi_{n}(q)$ denote the $n$-th cyclotomic polynomial in $q$. Recently,
Guo and Schlosser [Constr. Approx. 53 (2021), 155--200] put forward the
following conjecture: for an odd integer $n>1$, \begin{align*}
&\sum_{k=0}^{n-1}[8k-1]\frac{(q^{-1};q^4)_k^6(q^2;q^2)_{2k}}{(q^4;q^4)_k^6(q^{-1};q^2)_{2k}}q^{8k}\notag\\
&\quad\equiv\begin{cases}0 \pmod{[n]\Phi_n(q)^2}, &\text{if }n\equiv
1\pmod{4},\\[5pt] 0 \pmod{[n]},&\text{if }n\equiv 3\pmod{4}. \end{cases}
\end{align*} Applying the `creative microscoping' method and several summation
and transformation formulas for basic hypergeometric series and the Chinese
remainder theorem for coprime polynomials, we confirm the above conjecture, as
well as another similar $q$-supercongruence conjectured by Guo and Schlosser.
