I assume that this is the default exit tracking you are referring to? Not a custom "exit click" tracking... which means there isn't a lot of control you have over the rules / execution time.
Have you added any modifications to your exit links?
Or do you have all custom click tracking for everything? If you have custom rules, I suspect, like @Vinay_Chauhan that you must be triggering a JS error which is causing everything to break afterwards.
Is your site a SPA? If so, once the script breaks, everything will be broken until the page and the script is fully reloaded.
Would you be willing to share a link to your site (so that I can see the behaviour first-hand)? You can send it in a Direct/Private message, so that it's not available publicly here in this thread.