This is going to sound cold and heartless and out of context. But the reason we disciplined crying (not genuine, hurt, fear or sadness, but "fits") starting in toddlerhood, was to prevent this exact phenomenon. I have several friends who are awesome loving parents, but their kids cry all the time. The crying is either met with sympathy, attention, or eventually ignoring, but never a warning to nip the theatrics "or else" so to speak. It's very natural for sensitive kids to let their emotions run away easily. I personally feel it's what led to the uber-strict generations of "I'll give you something to cry about" parents of yore who were super tough on their kids so they wouldn't become "cry babies". That was excessive, but there's something to it. My husband and I took a more balanced approach that did include discipline for fits even though we are loving and understanding about crying when it warrants it.
Our friends surely thought we were a bit cold at times when our kids were crying and we said things like, "OK, that's ENOUGH" with a very warning tone, ready to enforce discipline if necessary. But we knew in those moments the situation did not warrant crying. Someone taking a toy from you, or not sharing with you, or getting a little boo boo is not license to melt down in our house unless there is exhaustion or hunger going on. We offered an initial piece of comfort or acknowledgment of what "happened" but then the kids needed to move on. Or else. We took cues from parents we know with resilient, happy kids, and they did not put up with the crying. Now our kids are tough and easy to bounce back too.
My two daughters are very emotional. My oldest (6) is sensitive and easily cries from "sensitive" things or boo boos. We go over our day and have conversations all the time about her feelings, she often talks through the "verge of tears", she needs her comforting blanket still, etc. My youngest (2 1/2) was born with a mega temper, and she would haul of and scream and rip people's hair out at the smallest infraction if allowed. I fear to think what her fits would be like by now if not nipped. Luckily, they both have excellent self control now and when the waterworks pop up they A) Control it, then B) we handle the issue. My girls are both VERY expressive and in full possession of their emotional spectrum. My son is less emotional in general but has excellent control of crying for boo boos.
Enforcing "no crying" at times with discipline is not squelching emotions, its teaching how to have self control when we would rather cry. It's one of life's most valuable lessons. The only other option is to wait for kids to outgrow it on their own with comforting etc, but that's not always the most effective as the habit is rooted. My cousin still calls home crying from college (and cries when professors give her bad grades!) and her life has been very easy and her parents were very loving.
I say continue being loving and attentive, but start to crack down on the crying after one calm warning if you have assessed that there is no bigger problem going on. When the waters are calm say, "Sweetie, you're now almost 6 and it's time to act like a young lady and not cry so much. The next time I ask you to calm down and you keep crying x will happen. I love you and I care about what is bothering you, but that behavior will not be allowed." Pick your most effective consequence and follow through. Also be sure to praise her sincerely when she does bounce back form a disappointment or injury without crying. Let her see how proud it makes her when she's "tough". My son loves to run around saying , "Hey, mom, I just fell off the counter and hit my lip (or whatever), but I'm TOUGH!" etc. Once she gets this under control, she'll actually be BETTER at expressing herself and understanding when something is serious or not.