Go to Bing.com.
On the new search results page, look for the “hamburger” (three horizontal lines in a block in the upper right area.)
On the new menu select Settings. This will have the cogwheel icon.
This should expand the menu with some new options. Select More.
Scroll down and look for the block marked “Results” and uncheck the box “Open links from search results in a new tab or window”
Click Save on the bottom bar that shows up.
That will put the new searches in the same tab.