Yet somehow regrettably the "turn off Web History" leads to the more brutally called "Deleting Web History" page. With the warning that "Deleting Web History from your Google Account will erase all items from your Web History and stop your Web History from being recorded in the future"! So think twice!
And when you are really sure, click the gear icon at the top of the page. A try shows that the gear icon actually does not work from that page. But don't give up so quickly! Keep trying other buttons next to the icon and behold - clicking your name does indeed open a menu with "Account settings". Which leads to the page slightly differently entitled "Account overview", where you don't see any "Products" as per instructions, but with minimum guesswork you figure out that now the "Products" are called "Services". Which has indeed a link to Web history with the scary option to removeit all, once and forever!
But ... never give up! A careful look at the top of the Web history page reveals a "Pause" button. Great - now we give it a try! So let's google "Italy"! And then do it in the separate browser - Safari when signed off. What do we see?
The first links to the Wikipedia page are identical. But the second are not! The signed in page gives "Map of Italy" whereas the unsigned Safari - "Italy Travel Information and Travel Guide" both from Lonely planet. Which one is more relevant? I'd the say the second one on Safari where I am logged out. If I were looking for the map of Italy, I'd enter it that way. Further going down reveals that both searches on "Italy" are not at all identical despite of the Web History being paused. And moreover, switching back the History on does not seem to produce any changes!