Combine onion and vinegar in a bowl; let marinate for 5 minutes. Drain, reserving the vinegar for another use, such as salad dressing.
Place 2 tortillas in a large nonstick skillet over medium-high heat (they will overlap); warm for about 45 seconds, then flip.
Pull the tortillas up the edges of the pan so they are no longer overlapping.
Working on one half of each tortilla, sprinkle one-fourth of the cheese, cover with 2 slices of turkey, 4 thin slices of red bell pepper, 3-4 basil leaves and top with one-fourth of the onion.
Fold the tortillas in half, press gently with a spatula to flatten and cook until the cheese starts to melt, about 2 minutes.
Flip and continue cooking until the second side is golden, 1 to 2 minutes more.
Transfer to a plate and cover with foil to keep warm.
Make two more quesadillas with the remaining ingredients. Serve warm.