Social media platforms are accused repeatedly of creating environments in which women are bullied and harassed. We argue that online aggression toward women aims to reinforce traditional feminine norms and stereotypes. In a mixed methods study, we find that this type of aggression on Twitter is common and extensive and that it can spread far beyond the original target. We locate over 2.9 million tweets in one week that contain instances of gendered insults (e.g., "bitch," "cunt," "slut," or "whore")-averaging 419,000 sexist slurs per day. The vast majority of these tweets are negative in sentiment. We analyze the social networks of the conversations that ensue in several cases and demonstrate how the use of "replies," "retweets," and "likes" can further victimize a target. Additionally, we develop a sentiment classifier that we use in a regression analysis to compare the negativity of sexist messages. We find that words in a message that reinforce feminine stereotypes inflate the negative sentiment of tweets to a significant and sizeable degree. These terms include those insulting someone's appearance (e.g., "ugly"), intellect (e.g., "stupid"), sexual experience (e.g., "promiscuous"), mental stability (e.g., "crazy"), and age ("old"). Messages enforcing beauty norms tend to be particularly negative. In sum, hostile, sexist tweets are strategic in nature. They aim to promote traditional, cultural beliefs about femininity, such as beauty ideals, and they shame victims by accusing them of falling short of these standards.