IRL, fighters do carry bombs, F-16 can serve as a ground support plane (same goes for F-15, and, in fact, every modern fighter), F/A-18 even has it in it's name. Dropping your bombs and engaging in a dogfight works quite well, as long as you don't get nailed before you make the pass (then, you can use emergency jettison to gain maneuverability, but you lose the bombs). Whether fighters are issued bombs is another thing, your squadron in WIH wouldn't have them even if Uhlans could technically carry them.
In-universe explanation would be that their targeting systems and interface rails weren't made for torpedoes.