Bart Simpson hasn't aged a day since The Simpsons premiered in 1989, which is interesting since he had his 10th birthday in season 3. Rather than age-up the character, the animators seem to have smoothed out some of Bart's wrinkles. As far as longstanding creative decisions go, this one is so obvious as to be unquestionable. After all, Bart's particular brand of anarchic, preteen rebellion is so tied to his maturity that his character traits would certainly read very differently at any other age (something that has been analyzed in non-canon Simpsons episodes).
Bart is hardly the only character that has been subjected to this strange limbo. On the contrary, every Simpsons character has long operated under the auspices of a floating timeline, including Mr. Burns. This has naturally created a number of discrepancies in newer episodes between the characters' assumed ages and the eras in which they lived. Principal Skinner and Grandpa, for example, have fought in the Vietnam War and World War II, respectively, which makes them both much older than they are meant to be in 2020. The writers have on occasion tried reconcile problems like these through retcons, such as when they moved Homer and Marge's courtship forward a couple of decades in "That '90s Show," but this technique tends to yield mixed reactions from audiences.
With so much fluctuation in the show's chronology, Simpsons' writers have only ever presented one proper, canonical birthday of Bart's: in "Radio Bart," when he officially turned 10 years old after having been 9 years old for the first few seasons of the show. The handling of Bart's birthday bears some resemblance to Marge and Grandpa's, whose birthdays (and other characters' inability to remember them) have often been used as the basis for jokes. However, Bart's 10th birthday has been treated with a greater level of reverence with regards to not being repeated, most likely because the relative impact of a birthday would be so much greater at his age.
Of course, knowing the specific date of Bart's birthday means that we can orient other events around it. The date has been given as both April 1 and February 23, although the latter originates from the non-canon "Simpsorama" and is often contradicted, so it should probably be disregarded. This means that the occasion has been danced around, if not mentioned, as in "So It's Come to This: A Simpsons Clip Show," which happens on April Fools' Day. His birthday was also explicitly shown in "Treehouse of Horror III" during the Clown Without Pity segment, but, as is the case with almost all of the show's Halloween episodes, this occurrence is considered non-canon.
On a show with such a conspicuous floating timeline, Bart having had only one 10th birthday might seem counter-intuitive to some viewers. However, the writers should probably be applauded for this, because, apart from perhaps Lisa becoming a vegetarian, The Simpsons hasn't had too many events that have had a long-lasting impact on the story. And yet, through any number of retcons, this is one particularly significant milestone that The Simpsons has never repeated or delegitimized.