It's not so much about whether you would use C# for web development, but rather, if you were using C#, you'd be using the .NET Framework, and from a web perspective, you'd be using ASP.NET.
You could also use ASP.NET, and all of the features of the .NET Framework, but "tie it all together" with VB.NET code if you wanted to.
So, it's not really so much about the "back-end" programming language that you use, you would probably pick whichever is most comfortable to you (especially in the .NET world where you effectively have a good choice), it's really more about the "front-end" scripts/languages that you would use.
For these, you really need (x)HTML, CSS & JavaScript. These 3 languages/scripts can be "laid on top of" virtually every other "back-end" technology that you would care to use (these are basically the things sent to the client's browser, so run locally on the client machine, rather than on your web server). Your back-end can now be ASP.NET, Java or any other platform you care to use, with the most "popular" ones being ASP.NET, Java (JSP) and PHP. (I'm sure I've missed many others out!)
Of course, on the server side of things, if you do need to run some intensive computations in response to user interaction, running pure C/C++ on the server will give you the best performance, you could even invoke these programs on the server using CGI, however, that's getting a bit old now and possibly has more security implications that other mechanisms.
One of the beauties of web development is that it's generally a "stack" of technologies working together to create the overall application, and in some cases you're free to pick and choose amongst technologies for each of the "layers" of the application. Stuff like ASP.NET (and the whole of the .NET world) attempts to "unify" these layers somewhat and give the developer a sort of "one-stop-shop" for creating web based applications.
Whilst writing web-based applications in C or C++ is (I think) relatively rare, it is certainly possible. I think you would have to play around with some different technologies for different layers of your stack to see what you're most comfortable with.
There doesn't seem to be a whole lot out there, tutorial-wise, for web based development with C/C++, but you could start by taking a look here:
C++ web development framework