Just as @mevrael already wrote, writing a website in C should only be considered after optimizing everything else and still needing more performance (Facebook, Google,....). There might be one more aspect: Crunching numbers. When doing heavy calculations on your backend, I highly recommend using a system language and outsourcing computations to GPUs via OpenCL or CUDA.
Personally, however, I would not go for C or even C++ any more. We are at the end of 2016 with new tech and languages which get you the C-speed, but with a lot more convenience on the developer-side. Personally, I would use Rust, as it prevents many error classes during compile-time. If there are no logical errors and you optimized your algorithms, you will have a rock-solid, high-performance application.
By the way, I don't believe that C will be used for WASM. I think, most WASM modules will be C++ or Java or Python or C#. In WASM, they will all have the very same performance, when given the same algorithm, so why should anyone go down the painful path of writing C?