"Funny" is only a minor and not actually necessary part of "make fun of". To "make fun of" is to make jokes about someone or something in an unkind way, to mock, tease, ridicule, laugh at, taunt, mimic, parody, deride, send up (British, informal), scoff at, sneer at, lampoon, make a fool of, pour scorn on, take the mickey out of (British, informal), take the piss out of (taboo & slang), satirize, pull someone's leg, hold up to ridicule, make a monkey of, make sport of, make the butt of, ...

All of which are ways of "making something sound bad". That's from a dictionary and thesaurus, but actual use, according to the Google hits that aren't dictionaries or thesauruses, agrees with them. Making something sound bad is the whole purpose of making fun of it. The "fun" part is the method of accomplishing that.