What you are looking for is a “stopover”.
Policies on stopovers are extremely variable. Some airlines, in cooperation with tourism boards of their home country. Actually, actually encourage stopovers in their home country/city (often with special deals on hotels or simplified formalities for visas).
That is (or was at some point) the case for Cathay Pacific in Hong Kong, Turkish Airlines in Istanbul, and at least one of the middle-eastern carriers (can’t remember which of Qatar Airlines, Emirates or Etihad), but there are probably many more.
Note that this is usually only valid for trips with a stop in their main hub, not in other locations.
Other airlines on the contrary will charge extra for stopovers and/or make it difficult to book stopovers.
If the website does not spontaneously suggest a stopover when you try to book a trip, the standard option is to use the “multi-city” option, which should try to find a matching round-trip fare allowing a stopover if one exists (otherwise it will indeed fall back to much less favourable fares).
If you want to find possible combinations (of airports, airlines and fares), you can use a tool like Expertflyer to list all fares from A to C and check the fare rules of each fare to see if stopovers are allowed, and if so, where, and at what cost.
