  Let $p>3$ be a prime, and let $m$ be an integer with $p\nmid m$. In the paper
we prove some supercongruences concerning $$\align
&\sum_{k=0}^{p-1}\frac{\binom{2k}k\binom{3k}k}{54^k},\
\sum_{k=0}^{p-1}\frac{\binom{2k}k\binom{4k}{2k}}{128^k},\
\sum_{k=0}^{p-1}\frac{\binom{3k}k\binom{6k}{3k}}{432^k},
  &\sum_{k=0}^{p-1}\frac{\binom{2k}k^2\binom{3k}{k}}{m^k},
\sum_{k=0}^{p-1}\frac{\binom{2k}k^2\binom{4k}{2k}}{m^k},\
\sum_{k=0}^{p-1}\f{\binom{2k}k\binom{3k}{k}\binom{6k}{3k}}{m^k}\mod
{p^2}.\endalign$$ Thus we solve some conjectures of Zhi-Wei Sun and the author.
