The WebBrowser control is not a silverlight object and sits on top of all content. You need to flip Visibility between the WebBrowser control and a rectangle with the fill set to WebBrowserBrush when the child window is visible and back when it's not. Something like:
<webbrowser x:name="browserControl" source="{Binding URL}" xmlns:x="#unknown" />
<rectangle x:name="browserControlRect" xmlns:x="#unknown">
<rectangle.fill>
<webbrowserbrush sourcename="browserControl" />
</rectangle.fill>
</rectangle>
Then either VSM, Behaviours, or code to do the toggling.