What word means what many people think 'ironic' means?

I'd say something about the matter, but I think I'll let George Carlin say it instead because he's more eloquent:

Irony deals with opposites; it has nothing to do with coincidence. If two baseball players from the same hometown, on different teams, receive the same uniform number, it is not ironic. It is a coincidence. If Barry Bonds attains lifetime statistics identical to his father's, it will not be ironic. It will be a coincidence. Irony is "a state of affairs that is the reverse of what was to be expected; a result opposite to and in mockery of the appropriate result." For instance:

If a diabetic, on his way to buy insulin, is killed by a runaway truck, he is the victim of an accident. If the truck was delivering sugar, he is the victim of an oddly poetic coincidence. But if the truck was delivering insulin, ah! Then he is the victim of an irony.

If a Kurd, after surviving a bloody battle with Saddam Hussein's army and a long, difficult escape through the mountains, is crushed and killed by a parachute drop of humanitarian aid, that, my friend, is irony writ large.

Darryl Stingley, the pro football player, was paralyzed after a brutal hit by Jack Tatum. Now Darryl Stingley's son plays football, and if the son should become paralyzed while playing, it will not be ironic. It will be coincidental. If Darryl Stingley's son paralyzes someone else, that will be closer to ironic. If he paralyzes Jack Tatum's son that will be precisely ironic.

Ironically, the word you're looking for is ironic, which according to Webster's means

happening in the opposite way to what is expected, and typically causing wry amusement because of this

Sarcasm is "the use of irony to mock or convey contempt," but it is not precisely the same thing.

If you mean the sense Alanis Morissette intended, then "unfortunate"?

Depending on the misuse, "coincidence", "apposite", or "unfortunate".