Harry Potter and the New Tournament
Radical Broom Racing
Rather than the Triwizard Tournament being hosted at Hogwarts, a series of smaller tournaments at Hogwarts takes place. These tournaments include: broom racing, Quidditch, academic knowledge, wizard chess, potion brewing, and dueling.
Harry and Ron try out for Quidditch and Dueling. Ron is pathetic in any position he tries and Harry doesn't make the Quidditch team because he lost the Snitch to Cedric in the Tryout. Ron gets into wizard chess (moderately skilled). Hermione gets into potions, academia, and dueling (excelling in all three). Ron and Hermione start hanging around new friends who are also in the same clubs, not really realizing that they're excluding Harry since he's become somewhat withdrawn due to his loss in the Quidditch Tryouts.
Shortly before the schools arrive, Harry goes for a spin on his Firebolt to vent his pent-up frustrations and loneliness. He draws attention from the Hogwarts racing team and they conscript him onto the team. Harry wins three of the Qualifying Races with some difficulty and gets allowed to race in the actual Tournament. Fleur also saw him flying and is quite impressed. At the Yule Ball, Harry doesn't manage to get a date and instead decides to take his broom to the dance. Fleur decides to 'dance' with Harry on her own broom, they quickly steal the spotlight. Afterwards, she promises to see him again…on the race track.
Miscellaneous Information
Based on the 1986 movie 'Rad' and certain ideas from 'Letters' by TheEndless7.
I've even got some ideas for how a race could be run and what could be considered fouls and penalties. The track itself is of course held up in the air. The racers need to fly through a series of floating rings (bronze, silver, and gold). Each ring is worth a specific amount of points (bronze-10, silver-25, gold-50) and the higher rings also come with increased levels of danger in attaining them.
There are random obstacles that line the race course like moving boulders, fireballs, Portkeys (these drag a racer further back down the track, causing them to lose time and points), and even Magical Creatures that attack if a racer gets too close. Other penalties include purposely knocking a racer off their broom, carrying and/or using one's wand during the race, or bewitching an opponent prior to the race. All of the aforementioned obstacles and penalties cause a racer to lose points should they be hit.
And finally there are two ways to win the race, which usually go hand-in-hand. The racer with the greatest amount of points and/or crossing the finish line first. Kinda like how catching a Snitch in Quidditch doesn't always mean that that team wins the game, i.e. Krum catching the Snitch at the World Tournament.