I am a big fan of Rust, but as it aims to be a better C++, it is probably a better fit for OS development, game engines, embedded systems, databases, complex desktop applications (Photoshop/Word/Chrome), etc. While Rust is quite expressive for a systems programming language, its banner features are the borrow checker (+ lifetimes, etc) and powerful static type system. Rust emphasizes zero-cost abstractions with compiler-enforced memory & thread safety. The popular web development languages are dynamically typed and interpreted, with an emphasis on rapid development, which is a very different niche than Rust claims to fill.
Perhaps Rust could be used to build a highly optimized server for a large site, but at that point it would probably better to write the application to run on the JVM. Although the JVM may be slower (within a constant factor, and assuming Rust is eventually as fast as C++) and more memory-intensive than the optimized native code produced by Rust's LLVM based compiler, the JVM's extensive tooling and run-time profiling tools make it a natural choice for high performance web servers.