"Thru" is in Dictionary.com, the definitive source for American usage. It's regarded as "informal" and "simplified," and as JR says it is therefore inappropriate for any formal writing. Nonetheless you'll see it everywhere that saving space or looking cool is a priority, such as advertising, webspeak, and the headlines of supermarket tabloids.
I question the "simplified" part. Yes it contains fewer letters, but does that make it easier for someone learning English to spell or read? I don't believe there are any other one-syllable words, or polysyllables with the accent on the last syllable, that end in the cardinal U vowel, in which the sound is spelled U. It's spelled in myriad ways, including OUGH, OO, UE and EW, but never U.
A possible exception is tabu, which is in the dictionary, but everyone except anthropologists spells it taboo. Even an unaccented syllable... all I can think of is ecru, which is a French word that only women know because men can only identify ten colors.

I don't think "thru" is a helpful neologism if we're not going to finish the job and respell the other OUGH words.
"Donut," on the other hand, is not considered an incorrect spelling, but "doughnut" is the preferred variant. It's an American word, coined about 200 years ago, so I guess we get to decide how it's spelled. It really is just what it looks like, a combination of "dough" and "nut," but I can't find an explanation for the coinage since there are no nuts in the batter and it doesn't look or taste like a nut. You probably see the word spelled out more often on neon signs than in any other medium, so it makes sense that they would go for the shorter version. If you're hungry for a donut you want to see that roadsign from as far away as possible, so reducing the number of letters almost by half and making them all larger makes it easier to spot.
The word "flu" has been around almost as long as "doughnut." "Influenza" is not only a mouthful, but it conforms to Spanish and Italian phonetic patterns rather than English, so it doesn't trip off the tongue lightly. (It's simply the word "influence" in those languages and don't ask me about the reasoning behind that.) When you start talking about the Spanish influenza or the swine influenza or the avian influenza, it's easy to see why you'd quickly lop off three syllables--even in writing! In fact, the secondary definition of "flu" is a specific strain of the virus such as bird flu instead of avian influenza.
All of these shortenings are doubtless influenced by newspaper typesetting. Every column-inch of reporting they can cut by eliminating "unnecessary" letters makes room to sell one more column-inch of advertising. In the 1930s the Chicago Tribune launched a campaign to "simplify" spelling. It was the Great Depression and saving pennies was important. They championed spellings like "nite." (Forgive me if I don't have the details 100% right but it was a Chicago newspaper at approximately that time.)
Edit: It just dawned on me that "nut" has more than one meaning. A doughnut is in the shape of a torus, just like the threaded type of
nut that you screw onto a bolt. Maybe that's where it came from. I don't think "doughring" or "doughcircle" or (goddess forbid) "doughtorus" would have caught on.