Place grated carrots in a large bowl.

Combine sugar, water, and vinegar in a saucepan and bring to a boil; cook until sugar is dissolved, about 5 minutes. Reduce heat to medium; add garlic, fish sauce, and curry paste to sugar mixture and simmer until flavors blend, 2 to 5 minutes. Remove from heat and cool dressing completely, about 30 minutes.

Stir lime juice and red pepper flakes into dressing and pour over carrots. Add peanuts and toss to coat. Refrigerate until chilled, at least 30 minutes.