Without going to great lengths (and potentially great inconvenience) to prevent it, web sites are routinely allowed to save cookies and other persistent state data on your machine. But there are a few things you can do to reduce the volume of this data and (perhaps more importantly) the ability for this data to be used to track you:
Uncheck "Preload Top Hit in the Background" in Preferences->Search
Uncheck "Allow Plug-ins" in Preferences->Security. Completely delete Plug-ins from your system unless you absolutely need them.
Set Allow WebGL to "Ask" for all web sites in Preferences->Security
Set "Cookies and website data" to "Allow from current website only" in Preferences->Privacy. You could set it to "Always block", but many websites will no longer function properly. "Allow from current website only" is a reasonable trade-of of privacy and convenience.
Set "Website tracking" to "Ask websites not to track me" in Preferences->Privacy (this will have limited effect with only some websites)
In Safari->General, set "Safari opens with" to "A new private window" and use private browsing as much as possible. Open new private windows and tabs for each website you visit and close those windows and tabs when you are done with each site. This will keep saved state sandboxed so there is no sharing of cookies or other saved state across windows or tabs, and no permanent storage of state to manually manage.
You can also run a browser or system extension to manage cookies and other saved state data more aggressively, for example by deleting all website data every time you quit the browser (and make sure to quit often).
The specific behavior you outlined may be due to having those sites in your Bookmarks or (more likely) your Favorites, or the browser re-loading favicons or simillar for frequently visited sites. Private browsing should help with this, but also in the Safari Bookmarks menu, uncheck the "Show Frequently Visited in Favorites" selection. Most importantly, NEVER "Show Favorites" from the Bookmarks menu, and NEVER set new windows or tabs to open with Favorites in Safari Preferences->General. Not only will this allow cookies and other state to be set by your Favorites, it will actually cause a whole bunch of sites to be visited in the background, giving them your IP address and other device-identifiers.
When you show Favorites, Safari actually makes two requests to each site in the background, one as macOS and one as iOS (presumably to get the apple-touch-icon for display).
If tracking is your main concern, there are numerous other defenses that you should implement (for example, tracker blockers and defenses against device fingerprinting and cross-device tracking). If this is the case, there are various Qs & As elsewhere on Stack Exchange. Or, modify your question to be specific about your concern.
