The following should produce equivalent output:

  pub fn popcnt(mut x: u32) -> u32 {
    x.count_ones()
  }

which gets compiled to:

  example::popcnt:
        popcnt  eax, edi
        ret

For large bit vectors, an AVX2 implementation can outperform POPCNT. See "Faster Population Counts Using AVX2 Instructions" at https://academic.oup.com/comjnl/article/61/1/111/3852071

32 bits is not large enough, and the code Rust produces is indeed comically bad.