All toddlers go trhough this stage, they play next to each other rather than together, and i have witnessed many a scrap at mums and tots, as two wee girls fight over the dolly pram or the push along trike....and boys too!
they have no concept of sharing, or taking turns, they want what they want now! and have no care for the other childs feelings, toddlers are totally egocwntric
was chatting to a mum at mums and tots last week, 21 month old DD stood with me , randomly poking other mums similar aged DD, then DD gave her a proper shove ! just took DD away , told her she was not to shove, distravted her with a toy, and carried on talking!
that is what toddlers do ! it is inherent, and had the other mum said DD was a bully or violent, for displaying normal toddler behaviour, i would have been more than a little p*ssed off
and FWIW, DD got scratched in the face by a similar aged boy, as he wanted the car she was playing with, c'est la vie !
i would either get used to the behaviour or stay in until your daughter is 4
the 'problem' you are tolerating is normal behaviour, you are making it a bigger deal than it ought to be IMHO