Okay here is the deal. I’m assuming you’re asking this because you don’t want your parents to see what you do on the internet on your phone, perhaps you’re afraid you’ll get “caught”. I’m also guessing they technically own the phone, and the service and you are privileged enough to have control of it.
So that being said, you should not use that phone for stuff your parents wouldn’t approve of. If you choose to do it anyway, then you are accepting the trade-off of the risk of being caught, and they have the ultimate control over that. So don’t get upset if you get busted.
If, however, you are doing something legitimate (shopping for mom’s birthday and you don’t want her to see where you bought the gift from, for example), then disregard the above.
Now then, depending on the browser you use, there are a few different things that track your usage. You have the “History” which is a list of sites you’ve visited, and you can delete that in the browser settings. You also have cookies and temp files, which you can also delete, although it may or may not be a separate process depending the app. THEN you still have auto-complete which fills in fields like search bars and URLs with search phrases or website addresses you’ve previously entered.
You gotta delete all those things in order to be completely incognito.
However… if you do this on your home wi-fi (and your parents pay for the internet), then your parents could get a log of all activity from the ISP, including URLs visited by you on the wifi. Nothing you can do to delete that, sorry. If you use 3G or 4G data to surf the web on your phone, the phone provider has those same records.
P.S: Ctrl + Shit + N while in Chrome will open a new Incognito browser window.