The Firefox browser is available for iOS or Android, free on both platforms. How you delete the browser history in each is a little different.
On iOS, tap the hamburger menu in the bottom center and select Settings. Scroll down to Clear Private Data, and on the next screen you can turn off collection of browser history (or data caching, cookies, and offline website data) entirely. Click the Clear Private Data link at the bottom to clear all of the above. Note in Settings there is also a toggle to Close Private Tabs, which shuts them all down when you leave the browser, should you be using such tabs o' stealth.
On Android, Firefox is back to the three-dot menu at upper right. Select History to see the list, and click CLEAR BROWSER HISTORY at bottom to nix them all from existence. If you click the menu and go to Clear private data, you get a more granular way to delete browsing history, search history, downloads, form history, cookies, cache, and more. If you go to Privacy, you get the option to clear the private data of your choice whenever you quit the browser.