Thanks for the recommendations, I somehow managed to do what I wanted. Basically I was using the search tweets functionnality to get tweets but then I had no idea how to check if the tweet had a media. It seems like some documentation pages of the twitter API website are not up to date.
There is indeed a "media" parameter (nothing about that on the API website), which is used when someone post a tweet with pictures directly from twitter. If the picture is shared in any other way than uploaded to twitter, the media parameter of the tweet is empty ... actually the paramater itself is not even displayed BUT the "urls" parameter contains a bunch of usefull information including the website where the pictures are added from.
So what I did is : Getting tweets, looping through them to check if each one of them have a media parameter (picture(s) uploaded to twitter directly), if they do not have a media I check the urls parameter to see if the website is "accepted", In my case I didn't wanted any tweets includding Youtube and Instagram links.
Then I can do whatever I want with all those tweets that match my conditions.