Ooooh ooooh ooooh, I know this one!! Short version: it's just a transition. The more you execute a normal routine and don't act like it's a big deal, the more quickly your kid will work through it.
The longer version: I swear, kids -- particularly highly-spirited sensitive ones -- can sense guilt like a wolf can smell fear, and they will just go right in for the kill. The thing is, it's not intentional: the more sensitive/needy the kid is, the more attuned they are to your emotions, so they pick up the signal from you that something's not right, there's something to worry about here, and that then ramps up their anxiety. The more stable and routine you are with the dropoffs, the less you let yourself get tugged by guilt or sadness, the more your child will feed off your belief that this is a normal -- even exciting -- part of life. (Once my kid adjusted, she LOVED daycare, because they had WAY better toys than we did at home)
I learned this totally by accident. My emotional-canary-in-a-coal-mine DD had huge meltdowns at dropoff, and it freaking killed me, even though the teacher always told me that she was just fine when I left. So I'd give her extra hugs, repeatedly tell her I'd be back soon and she'd have lots of fun, and a whole bunch of stuff to try to reassure her and calm her down. Then one day I dropped her off, walked out of the room to her panicky shrieks (with tears in my own eyes), and stopped outside the door to do something (tie my shoe? check email?). I swear, within 30 seconds, she was totally calm and happily playing. No exaggeration here: it was well under one minute. That's when it clicked in that my inconsistency and worry was actually hurting her and making the transition harder, not easier. So I stopped feeling guilty, began consistent three-hugs-and-leave dropoffs, and very shortly she stopped squalling at all when I dropped her off.
Meanwhile, same kid was still freaking out when my mom dropped her off a good two years later. Why? Because my mom felt terribly guilty and would go back for just one more hug, and then another. . . .
I know how hard it is to walk away when your kid is sobbing like she's never going to see you again. But the more quickly you can get over your own guilt and deal with this as just a normal part of life, the better it actually is for her.