So Caitlyn is strong when she can abuse her range advantage.
She strong in the early laning phase because she can poke her lane opponent and get a good lead in lane.
She's weak in early teamfights (mid game) because she doesn't bring out the same level of sustained damage that most other ADCs do. She's easily outdamaged by most ADCs.
She's strong late game because it gets to the point that a dead ADC = a lost teamfight. Caitlyn's range lets her hit higher priority targets without messing up her positioning. By late game, a 6 item ADC is also going to tear through anyone, so she doesn't have "less damage" anymore. Her traps also are great for sieging or defending a siege, or contesting an objective.
So yea, you're obviously going to lose to a Jinx in a 1v1. Caitlyn wins lane in that scenario, but her strong late game isn't because she outscales - it's because she has a better teamfight and objective control. On a strict damage basis, Caitlyn is still outclassed by most ADCs in the late game (she just scales well late game because she can apply that lower damage more easily / consistently / on better targets).