I've confirmed that this is definitely the Kagi extension causing this issue. Turning off the extension and using one of the built in search engines doesn't suffer this problem. Lastly, I know iOS 18 is a beta but figured i'd submit a bug for visibility before its fully released in the fall.
In Safari on iOS 18 Beta (beta 2 and beta 3 thus far - didn't test in beta 1). To reproduce, in Safari on iOS with the kagi extension enabled and logged in, open a new tab and go to a website such as reddit.com and let the page load. Tap the address bar and enter a search term and hit "go". The address bar and keyboard close but safari stays on the same page. If you click the address bar again OR wait a few seconds, the kagi search will begin executing and redirect. Note, clicking the address bar will bring it up with the keyboard like its about to execute and then immediately dismiss itself.
Every time I search on an existing tab/window the search executes.