One trap at level 3 is literally a forced back if not an actual 100-0 even without the support's damage. The CC just makes it possible. One W leads to a total of five autos, two or three of which are empowered depending on the state of the passive stacks, and a Q, all in the frame of two auto attacks, and the first auto in this example is actually while the trap is arming, so, yeah, any hard CC is an instant death against a Caitlyn who is positioned properly and ready to capitalize, with the exception of Aftershock users, who will still be forced to recall if they don't have potions. In the off chance the fast Cait combo doesn't 100-0 or force a back because you actually messed it up completely, you're still using your net to escape as you do it, so you're completely safe, too. Fun stuff. Literally the hardest part of playing Caitlyn in lane is managing mana, and that's not even that hard.
Seriously, I forget the exact math, but I was doing over 400 damage a combo against 100 armor practice dummies, so very nearly 700 actual damage in one combo after armor. In fact, not taking into account armor runes, Caitlyn can 100-0 most ADCs in the game off of one trap + ignite AT LEAST from level 3 up until level 8 with nothing but a Doran's blade. I forgot to account for armor runes when I tested and I stopped testing after 8 because I was disgusted, but that's something most ADCs need to pretend to have items for. Now, obviously you're not going to be ready for EVERY trap, but one CC into a trap is just instant death after first back for sure. It's practically free. It's also ridiculously easy to hit a full Caitlyn fast shot combo for some reason.
TL;DR stupid math and mechanics, just take cleanse and pray to whatever deity you believe in
Also, not sure if this still works this way, but I remember that while Morgana black shield stops traps from rooting, it doesn't stop the Headshot mark. Haven't seen Morg against a Cait in a season or so because it usually gets picked FOR her instead.