When you enable "Request Desktop Site" all the browser does is change the
user agent
string provided as part of the
request header
.
If you website has a different behaviour when this is enabled it's because your website or its components are designed change behaviour for mobile devices.
So look at your website and it's components and try and disable the behaviour.
Default user agent:
Mozilla/5.0 (iPhone; U; CPU iPhone OS 5_1_1 like Mac OS X; en-gb)
AppleWebKit/534.46.0 (KHTML, like Gecko)
CriOS/19.0.1084.60 Mobile/9B206 Safari/7534.48.3
Request Desktop user agent:
Mozilla/5.0 (Macintosh; Intel Mac OS X 10_7_3)
AppleWebKit/534.53.11 (KHTML, like Gecko)
Version/5.1.3 Safari/534.53.10
That's the only different from the user end point perspective. Everything else the behaviour of the website.