Why Is My Two Year Old Being So Mean Lately?

Hey, i need some advice. my daughter is almost 2 years old. and i don't know if this is a phase that she is going through. but she's so naughty lately. she's acting like a spoiled brat, and being a bully. she screams and screams when i won't give her some thing she wants. and then she gets violent and takes it out on other people. like my 6 month old son. she will some times take a toy and hit him in the head with it. or to another one year old i babysit, she will shove him to the ground hard, or try to hit him with stuff. and sometimes she'll even hit herself with things when she's really mad. i don't know what to do. i try to do every thing i can. i scold her, tell her it was wrong, put her on time outs. but she doesn't listen still. she cries a little, but it doesn't help, because an hour later she'll be trying to do it again. i feel like such a bad mother, feel like i'm scolding her all day. i don't think i ever did anything to spoil her, i don't give in to her tantrums. is her behavior normal toddler stuff, or is it indicating that she may grow up and be a spoiled brat who bullies others. i dunno. i just need some advice. and please don't judge. i'm trying my best. any advice for how to get her to start behaving and being nicer to other children?