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The first place to look is Apple's own iPad User Guide for iOS 7 which explains the new search method, both for the Internet and on the page being viewed. Here's an excerpt from page 46.
IPad User Guide for iOS 7 (Credit: Apple)
What happens in iOS 7, when you tap the integrated URL/Search bar at the top, is that you're instantly taken to a Favorites page. That shift in context, away from the page you're viewing, is what may cause confusion. In this example, I am visiting www.macobserver.com, and after tapping the search field, Safari looks like this:
My default favorites.
How is that Favorites page defined? Apple's User Guide explains: “Have your favorites top the list. Select them at Settings -> Safari -> Favorites.
It does not appear that there is any way to stay on the page being searched because, apparently, with an integrated URL/Search field, Apple has to assume that you're searching the Internet, not the page. In my case, here are my (only) options:
Note the caption above. “Quickly access Favorite bookmarks when you … search”
Just tap on the search field and start typing the search term. For example, even though it isn't being displayed, I'm searching the home page of The Mac Observer for the term “Gamet.” At the bottom of this list is a gray bar with the notation: “On this page.” If you tap the search term below, Safari will search for the first occurrence of that term on the page you were viewing.
In my example here, it found Jeff Gamet's name highlighted in yellow below. At last, Safari is back to the page you're searching.
The first occurrence is highlighted.
Finally, at the bottom of the page being searched, there is a search bar that allows you to step through the next/previous occurrences or change the search term.
You can change the search term here.
Apple could have chosen to do this in several different ways, but taking the user away from the page being searched before the user expresses the search intention is what's apparently causing confusion. As with many new features in iOS 7, a little bit of practice and familiarity really helps.