As the last few years have seen an increase in online hostility and polarization both, we need to move beyond the fack-checking reflex or the praise for better moderation on social networking sites (SNS) and investigate their impact on social structures and social cohesion. In particular, the role of recommender systems deployed at large scale by digital platforms such as Facebook or Twitter has been overlooked. This paper draws on the literature on cognitive science, digital media, and opinion dynamics to propose a faithful replica of the entanglement between recommender systems, opinion dynamics and users' cognitive biais on SNSs like Twitter that is calibrated over a large scale longitudinal database of tweets from political activists. This model makes it possible to compare the consequences of various recommendation algorithms on the social fabric and to quantify their interaction with some major cognitive bias. In particular, we demonstrate that the recommender systems that seek to solely maximize users' engagement necessarily lead to an overexposure of users to negative content (up to 300\% for some of them), a phenomenon called algorithmic negativity bias, to a polarization of the opinion landscape, and to a concentration of social power in the hands of the most toxic users. The latter are more than twice as numerous in the top 1\% of the most influential users than in the overall population. Overall, our findings highlight the urgency to identify harmful implementations of recommender systems to individuals and society in order better regulate their deployment on systemic SNSs.