On My android I had the same goal a few month ago. Basic is in the URL and the Web Browser you use. The web site will identify the browser has "Safary XX M" M for Mobile device. So you get redirected to a url like for fake example, "www.google.com/m" Some browser you can just edit the m out but it's rare now. There are add-ons you can use to make the website believe your using an other version of the browser. Some browser have the feature build in like firefox, click on the option menu and there's a box to request the desktop site. But by default it always goes back to mobile after a restart. I also played with the about:config in a few browser and it never stay permanent but it can be done like this also.
I don't know safari much, but the principle should be the same. There are probably a few other solution out there. But then it got more complicated and I just gave up on those website on my mobile browser. Most want you to use the App cause it'S harder to block advertising and hit there revenue.
Do you mean you want to view webpages in the same size class as on a larger screen?
Well, it depends how the site is coded actually, but if you hold down on the "refresh" button, it'll give you an option to "request desktop site". But for some sites it doesn't work as per the way they've been written.