How to enter a URL into a smartphone without doing a search? Scenario: I'm talking in-person with someone and during the course of conversation I think of a website whose URL I know from memory, and I want them to type the URL into their smartphone and access the website directly without performing a web search (just like you could do on a traditional computer web browser in the address bar).

To be clear: I don't want it to do "search as you type", I don't want it to create suggestions based on partial matches, and I don't want it to do any kind of search that delivers search results. I don't want the entered string to be sent to a search engine to do any processing of any kind. I just want it to open a connection to the indicated website (i.e., go directly to DNS lookup followed by HTTP or HTTPS connection to the entered URL). Also, in this scenario, I have no way to generate a QR code for them (I don't own a smartphone, and might not be carrying my computer with me).

Do smartphones have that facility (i.e., a field to enter a URL without passing it to a search engine, exactly like the old browsers used to do with the address bar)? If so, how do I instruct the Android user (and/or iOS user) to find the field where they can type in the URL and not have it be processed by a search engine?
These days, browsers on mobile devices, use one bar for searching with an search engine and inserting a specific URL request of web site. 
What you need to do is simple tell them "Touch top bar/address bar and insert this specific URL" or "Insert this URL in the bar and send".
You need to know that if the inserted text is a correctly formatted URL (protocol://domainOrIPAddress:port/path/filename) then the browser will show you directly the web page, otherwise it will use the default search engine for retrieving a possible list of sites that you maybe want to see for the inserted keyword.

In summary, you don't need something special, everything should work without you doing anything on browser settings. However I think there is an option to deactivate the bar for searching cases and only use it for inserting URL, but it's more convenient how it works normally, so by default is one bar for searching and URL request in all browsers.

However, for not letting search engines look on your search/request you need to uncheck some settings on browsers like "Suggestions by Search Engine/Bookmarks".