Most(?) browser vulnerabilities are caused by errors in C++ code which would not be exploitable in memory safe languages. One of the goals of Mozilla's Servo is to write a browser that's memory safe without compromising performance.
I think Servo's "safety" is ultimately due to the fact that it's built on Rust. Rust, however, seems to be ultimately built on C, unless I'm mistaken (having a hard time telling by briefly glancing through their Github, but it looks that way).
My point was that it's not a C specific problem, though. Most browsers are in fact built on C, I agree. This is due primarily to the speed and performance of the language that is harder to reach with other languages.
It is definitely a more difficult language to write, as it is much more "raw," but that doesn't make it inherently unsafe to use, or any more unsafe than other languages.
