How do you do a web search on an IPhone?
You can just swipe down on any home screen to reveal the search field at top of screen. Enter your search criteria and you'll see a range of results, including suggested websites.

If none of the results suit you, just swipe up and tap on the Search Web option that appears at the bottom of screen. This will do a search for you using Google (or whatever search engine you have set as your default) in a new Safari tab.

NOTES:


It may be worth you checking whether you've configured Spotlight Suggestions. You can do this via Settings > General > Spotlight Search.
You can also check your Safari Search settings within Settings > Safari.