Unlike Safari on the OS X and for iPad, there is no tab browsing for the iPhone version of Mobile Safari. This is understandable due to the small screen size of iPhone. Instead you open links in a separate page, somewhat similar to a browser window in Safari for OS X.
To open an URL link on a new page, tap and hold on the link and select Open in New Page from the pop up prompt. This will move the current page out of view and open a new page with the URL.
You might not like this default behaviour but would rather open new pages in the background. This will allow you to continue reading the current page, but still save interesting links in background pages.
To change the behaviour and open URL link in background, go to Settings > Safari > Open Links and select the In Background option.
When you next tap and hold on an URL link in Mobile Safari, the pop up prompt selection is changed to display Open in Background.
When you open an URL in the background, the current page remains on the screen for you to continue reading. Tap on the Pages icon at right corner of Safari toolbar to access those background pages that you created.