I am using Windows NT 4.0 and the built-in web browser Internet Explorer 2.0 seems to have a problem locating most of the web sites. It looks like the browser is not using HTTP/1.1 and doesn't pass the Host header to the server. This causes the server to response with 403 or other error. Or at least this is my current hypothesis of the situation. How can I enable HTTP/1.1. According to Wikipedia it should be supported.
It is not possible.
HTTP/1.1 was ratified in RFC 2616 in June, 1999. Internet Explorer 2.0 was released on November, 22 1995; three and half years before HTTP/1.1 was invented. As such, it knows nothing about it and thus cannot support it.
Your best bet is simply to update to a newer version of IE. Service-pack 6 for NT4 includes IE 6.0 SP1. You should apply the latest/last service-pack (SP6a) to NT4 anyway, so getting HTTP/1.1 through an IE upgrade is just another benefit of doing so.