Summarize:

Tap the Settings app icon, which resembles a grey box with gears on it. This process will also work on an iPad or an iPod Touch. It's roughly a third of the way down the Settings page. This option is near the bottom of the page. It's at the bottom of the screen. Doing so will clear your Safari cookies and other web data. This option removes your on-iPhone search history as well. If you just want to delete cookies, instead tap Advanced at the bottom of the page, tap Website Data, tap Remove All Website Data, and tap Remove Now.
Open your iPhone's  Settings. Scroll down and tap Safari. Scroll down and tap Clear History and Website Data. Tap Clear History and Data when prompted.