

You could try Incoquito - it's a paid app and allows you to either automatically close all incognito tabs when the screen turns off or to disable incognito browsing altogether by preventing incognito tabs from being opened at all. This uses the officially-supported way that device manufacturers use in order to control the starting page and default bookmarks of Chrome on their devices. This installs a content provider which, among other things, allows disabling Incognito mode in Chrome.

If you have rooted your phone, you can build and push a system ChromeCustomizations app.
