Using your browser's private or incognito mode prevents Bing from remembering your search history, but it's easy enough to manually clear your search engine's history even after browsing in its normal mode. You can do that from the site's Options page.
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Open Bing in your browser and then open the search engine's settings by clicking the gear icon in the upper-right corner of the Bing page.
Switch to the History tab and click Turn off to stop Bing from saving your future searches.
You can erase individual search results or all of them at once.
To erase a single result, place the mouse pointer over an item in the history and click Clear when it appears. If you don't see the item you want to erase, search for it using the search box on the page or click All dates to narrow the date range.
To erase all searches from your history, click Clear all. Then choose Yes, clear all search history to confirm you want to erase your entire history.
Optional: If you want to hide the automatic suggestions that pop up alongside your search history, go to Options and click the Settings tab. Then clear the checkbox for Turn suggestions on and click Save. If you like the suggestions, you don't need to turn them off in order to delete your history.