Originally the web was written in Perl. Then PHP was written in Perl and the web started being written in PHP. Since then Perl has acquired better web page writing tools (see CPAN) and PHP has gotten more and longer functions names. Yes, you can write server side code in Perl - assuming that is what you mean by "design web pages by using perl". BTW, PerlMonks is written in Perl - which currently may not be the best advertisement for Perl scripted sites :(.

.NET is designed as a support tool (framework) for writing Windows applications. A Windows box can be used as a web server so .NET can be used for writing server side code. However most server side code only really does two things - it processes text and talks to databases. Perl is excellent for both those tasks. .NET, not so much. Note that .NET is not a language, it is a framework, so you need to learn the framework and at least one language to use it and the framework is likely to be the harder part!