Hi, S.. Well, kids today do have a much different world to cope with than you and I did growing up. They lose their sheltered innocence at a younger age because the violence that you and I could put at a distance (the Vietnam War, for example), is happenning in their school. Even if we turn off the TV, there are WAY more pressures in school than most of us mature adults had to deal with when we were kids.
So it's understandable if our kids seem to be more afraid and react more strongly than we are used to seeing.
I don't know what your family's spiritual beliefs are, but if you folks do believe in God, I would start emphasizing His love, His protection, His power, as antidotes for your daughter's fear (and yours, too). In my life, God's presence and protection are the only things that help my own anxiety.
The reality is that we parents cannot shield our kids from every bad thing in the world, and the kids know it. What they need to know, I think, is that there is someone who can be with them every second of the day and night, helping them through every hard situation. God won't take away all the difficulties, but He has promise to be there, guiding and making our paths straight. To me, this gives me the courage to get out of bed in the morning and walk through my life at all. It also gives me the peace to lie down at night and sleep. I have very few nightmares, even though I have been through lots of trauma, and my life is fairly difficult.
It is also quite expected for children to experience anxiety that they can't define when they change schools, approach puberty, and so on. At age 11, her body is about to go through changes, if they haven't started already. That can do a real number on a child's head and heart even if that child is happy about entering puberty.
Dreams are the mind's way of trying to make sense out of what's happenning, and most of what goes on in dreams is symbolic and can be important in helping a child to deal with fears. See if you can get your daughter to talk to you about what happens in her dreams. She may be old enough to start a dream journal; that may be one way for her to take on a sense of power in this area instead of being a victim of scary dreams.
I do hope my input is helpful to you, and I will pray for your daughter's fears to subside. Take a look at your own fears, too, which you may accidentally be passing on to her (I'm not judging you; we all do this).
God bless you with peace,