It's compilers and compiler optimizations that make code run fast. The real question is if the Rust language and the richer memory semantics it has help the Rust compiler to provide a bit more context for optimizing that the C compiler wouldn't have do unless you hand optimize your code.

If you do hand optimize your code, all bets are off. With both languages. But I think the notion that the Rust compiler has more context for optimizing than the C compiler is maybe not as controversial as the notion that language X is better/faster than language Y. Ultimately, producing fast/optimal code in C kind of is the whole point of C. And there aren't really any hacks you can do in C that you can't do in Rust, or vice versa. So, it would be hard to make the case that Rust is slower than C or the other way around.

However, there have been a few rewrites of popular unix tools in Rust that benchmark a bit faster than their C equivalents. Could those be optimized in C. Probably; but they just haven't. But there is a case there of arguing that maybe Rust code is a bit easier to make fast than C code.