Hi J.,
Thank you so much for posting this, and thanks to all the responses, which I read with interest. My six year old has started this same kind of behavior.
In general, I don't doubt that her self esteem is pretty good, but I think the academic pressure of kindergarten (I can't believe there is any, but at least at our school, there is) has meant that she has to deal with the idea of being evaluated and compared to other kids. I wish this were not happening at her school , but I think it is the reality of schooling. And she is an only child so she is not used to being compared, even though she did have years of preschool. I think she is struggling with an environment where she is just one of many, and being part of the group is a lesson she has to learn.
Sometimes its clear she is talking this way to manipulate, as it often comes up when she is getting scolded for something. Its like she is going to beat us to the punch by punishing herself. She'll say, "you're right, I'm stupid." I've tried to do what some of the other respondents said -- tell her she is not stupid but that I did not like X behavior, and then try to move on and not play into it too much.
But some days I believe she is genuinely feeling bad about herself when she doesn't perform as well as another kid at this task or that. I think this is a life lesson too, although I don't always feel confident about how to help her learn it. I just remind her that everyone needs help sometimes, that new skills take practice, and so on, and keep encouraging her in the activities in which she is involved.
Anyway, just know that I'm with you! I'm especially concerned with a daughter, as we know that there are a million things out there in our culture waiting to convince girls that they lack value. I trust that our daughters have much more support and opportunity than my generation did, however, and that we can model self-esteem for them.
Best,
S.