How do I stop Google pop-ups in Safari on iPhone SE?

Every time I use safari on my iPhone using Safari I get a pop up from Google and I have to select stay logged out. If I want to use Google and their app I would have done so. It very annoying but I don’t know how to get it to stop. Someone told me to stop using private browsing and it will stop. First, it did not stop. I get the same annoying pop up from Google whether I am in private browsing or not. Second, I’m a little concerned that it does pop up on private browsing. I thought it was PRIVATE but it doesn’t feel like it is private at all if Google knows and sends me a pop up every time I type a question into the Safari browser. Can someone please tell me how to stop this preferable stopped in both private and non-private would be nice but I’d settle for just one option with out the annoying pop ups? I’m using an iPhone SE (2022) running iOS 18.3.2. Thanks.

[Re-Titled by Moderator]