A couple of questions:
1. Do you have tabbed browsing enabled and is it the tabs that are closing, so that only one tab is open at a time?
2. Is the new website (Bwebsite) overwriting Awebsite or is Bwebsite opening a new tab and closing Awebsite?
If tabbed browsing is not working, just check that it is enabled by going to:
Tools, internet options, general tab, and selecting 'settings' next to where it says tabs. Below is a pic with my default settings.