This is up to the website developer for each website to include a mobile version, and they are doing a favor for most cell phone browsers.
Since Safari on the iPhone is a mobile version of Safari, it is detected as such when visiting a website that also provides a mobile version of the website. Most websites that also provide a mobile version include a link to go to the full version. If so, select the link to go to the full version and save the URL as a bookmark and/or as a webclip on your home screen. When selecting the webclip for the full version of the website, you will be taken to the full version.