It presumably was made around the same time as Fix-It Felix, Jr., due to it being set up next to Fix-It Felix, Jr. when the arcade first opened 30 years ago. TurboTime was once one of the most popular games in Litwak's Arcade, and Turbo loved the attention he received.
When Mr. Litwak installed a new racing game, RoadBlasters, it stole Turbo's "thunder", causing him to become jealous. As a result, he left his game and tried to take over the new one, appearing on the screen while some kids were playing. He crashed into the car the kids were controlling, making the game crash. Instead of getting Turbo his attention and glory back as intended, this action caused both machines to be considered "out of order" and unplugged for good and Turbo was presumed to be dead. Since then, the characters of the arcade usually refer to a character game jumping during arcade hours as "going Turbo", and is something to be avoided. However, near the end of the film, it's revealed that Turbo had survived his game being unplugged, and had, disguised as King Candy, taken over Sugar Rush.
Like the other games created for the film, a TurboTime game was produced for the Wreck-It Ralph app.
TurboTime was placed next to Fix-It Felix, Jr., which explains the fact that of all the characters in the film, Felix was the only one who knows who Turbo really is and what the phrase "Going Turbo" actually meant and why King Candy was the only one who recognized Ralphat sight.
Turbo most likely "went Turbo" in 1987, since that was the year RoadBlasters was first released in arcades.
It's never explained what happened to the other characters from this game (including the "Turbo Twins", Turbo's rivals) after the game was unplugged, they were either rendered homeless like Q*bert and stranded in Game Central Station, or that they were all killed when they all couldn't evacuate their game in time. Since we never see them in Game Central Station, they were all most likely killed.
TurboTime is supposed to be a parody of Rally-X, but is portrayed as a racing game instead of a maze game (it has a steering wheel instead of a joystick, and its gameplay, showing an oval racetrack from above, only appears for a split-second) and also seems like a combination of Pole Position and Death Race.