These are not retweets. What I'm seeing is tweets of people who follow people I follow. For example if I follow John Doe I'll see a tweet from someone who follows John Doe too. At the top of the tweet it tells me "John Doe follows". How can I get rid if this?
Are you sure you're not seeing favourite tweets? This behaviour was implemented in 2014:
You can verify this by checking to see if there is a gold star visible on the tweet itself. As far as I am aware, there is not currently a way to disable this globally. You should be able to do it on an individual basis, similarly to disabling retweets from a specific person.