Certain Search Engines (Google for example) will retain your search history forever, regardless of any "Private Browsing" settings in Safari or any other browser product.
Please learn what Private Browsing is, and isn't; what it does, and what it does not: Turn Private Browsing on or off on your iPhone – Apple Support
Salient excerpt (with emphasis):
With Private Browsing, Safari won’t remember the pages you visit, your search history or your AutoFill information.
That's it. There is nothing more to Private Browsing than that.
To answer your question those suggestions are not arising from Safari itself.
Does anyone know what I can do about it?
Using a different Search Engine — one that does not retain your search history (Internet browsing activity, location, personal preferences... etc) — will help, but bear in mind everything you do on the Internet is subject to being intercepted, scrutinized, retained, subpoenaed and can be used as evidence against you in a court of law.
It should also go without saying that if you value your privacy or have any reasonable expectation of privacy do not use Google, or any Google product, for anything.
References:
Keep your browsing history private in Safari and Maps – Apple Support
https://duckduckgo.com/download/Private_Browsing.pdf
Google settles $5bn lawsuit for 'private mode' tracking
https://spreadprivacy.com/is-private-browsing-really-private/
potentially related:
Why Safari private browsing is retaining my history & website data on iPad? - Apple Community