Place a large rimmed baking sheet on center rack of oven; preheat the oven and sheet to 450°F.
Toss potatoes, artichokes, 3 Tbsp. oil, 1/2 tsp. salt, and 1/4 tsp. pepper in a large bowl. Arrange in an even layer on preheated sheet and roast, tossing once halfway through, until nicely browned, about 18 minutes.
Toss green beans, olives, 1 Tbsp. oil, and 1/4 tsp. salt in same bowl.
Season fish on both sides with 1/2 tsp. salt and 1/4 tsp. pepper, then coat with remaining 1 Tbsp. oil.
Remove baking sheet from oven and push potatoes and artichokes to one side, taking up about a third of pan. Place fish on another third, then bean mixture on remaining third. Roast until fish is cooked through, about 8 minutes.
Meanwhile, whisk mayonnaise, lemon zest, lemon juice, and remaining 1/4 tsp. salt and 1/2 tsp. pepper in a small bowl.
Flake halibut into pieces and transfer to a platter with all of the other roasted vegetables. Top with basil leaves and serve alongside lemon dressing.