The name does appear to be "search bar."
Cf.
www.mozilla.com/firefox/search.html:
"Products | Mozilla
The Firefox Search bar comes pre-loaded with Google, Yahoo, Amazon, eBay, Answers.com, ... It's easy to add more search engines to the Firefox Search bar. ..."
The name for the place where the URL goes is the "URL Text Field"
Cf.
http://www.laubach-on.ca/browser_screen.htm :
"URL Text Field: this field provides the address (universal resource locator) of the web site you are currently visiting."