This app was mentioned in 8 comments, with an average of 2.50 upvotes
Install Inbrowser. No, it's not an incest browser, just a coincidence that's the topic at hand...
https://play.google.com/store/apps/details?id=nu.tommie.inbrowser&hl=en_US
It completely kills the app upon screen lock or closing the app. It doesn't track anything from what I can tell.
Do you want this feature because the all the new tabs use up too much ram? If so, try Naked Browser, it uses minimal ram. You have options to limit the number of tabs cached, and even to 'forget' the tabs when you exit the browser, so next time you open it, all the previous tabs are closed.
Also, some private browsers have a setting to clear history every time you leave the app, so the old tabs will be closed. First one I found was this. I remember UC mini has this feature.
If you really want this feature, try some old apps, the new/popular ones all probably have the tabbing feature.
I use this one: https://play.google.com/store/apps/details?id=nu.tommie.inbrowser&hl=en
However, husband suggested just now that it may have to do with the keyboard as well. This one is open-source and doesn't record any strokes (unless you add to dictionary): https://play.google.com/store/apps/details?id=org.pocketworkstation.pckeyboard&hl=en
(Don't be let the app's name scare you)
To make things easier for my Android brethren (and sistren) give InBrowser - just fully automates incognito mode and as soon as you switch apps press "home" it disappears from recent apps.
I do a lot of gift shopping on my phone.
I use PIA's "inbrowser". Works for me every time.
https://play.google.com/store/apps/details?id=nu.tommie.inbrowser