I'm looking for a web browser that will not let the user access any
web site unless the parent first authorizes access to that site.

ChiBrow is the browser you're looking for:
http://www.chibrow.com

You, as the parent, have complete control over what websites your
child can visit as this quote from their website states:

"ChiBrow is a browser, not a filtering software per se. ChiBrow
protects children simply by giving parents complete control.  Parents
define what is appropriate for their children by determining what Web
sites they can access.  They can create or build on an existing list
of safe Web sites (Safe-Sites List), restrict access to a Safe-Sites
Search Engine, and/or open access to sites based on domains realms
(.edu, .gov and .org).   Because parents have complete control over
the browser, and hence the content that is displayed, they can be
certain that sites they allow their children to visit meet their
safety standards."
http://www.chibrow.com/about/

There are many other great features about this browser so I encourage
you to fully review their website.  I think you'll find that it's
exactly what you're looking for.  The yearly license for the browser
is $29.95.

Hope this helps you.  Please feel free to ask for clarification if
necessary and if you're satified with the answer, please rate it.
Thanks!
  
xemion-ga

Perfect!