While you can certainly limit "support" for a subset of web clients, you ought to be able to make it work for, at least, all modern web browsers to a greater or lesser extent. The difference really has more to do with the web browser than the web site. As far as dynamic behavior, I'd suggest going with a javascript framework that already addresses differences between major browsers. CSS is more problematic, but you should still be able to approximate the same look and feel, though it may take some work. Without knowing what specific problems you're having it's hard to know how to suggest that you solve them. Generally, I would target and test IE 8/7, FireFox, and Safari (or Chrome). You might want to throw in Opera as well, though my experience has been that if you get FF, you generally have already gotten it to work for Opera.
If you are having trouble with getting the generated HTML to work the way you want it to, you might want to try using ASP.NET MVC. It gives you much more control over your HTML. This can be really valuable in making things work the way you want it, though, WebForms code can obviously be made to work as well. Sometimes, though, you have to jump through a few hoops to get it to behave exactly the way you want because the output of web controls isn't always what you and I would produce in every situation.