I have watched and heard some rumours on Joker of Suicide Squad (2016) about him being Robin. Was Joker really Robin? Because Joker was not that guy we know.
The rumor/fan-fiction was
Second Robin Jason Todd was not killed by the psychotic villain after all but instead turned into him. Fans have cited ‘evidence’ in marketing material released during the run-up to Suicide Squad’s, claiming that The Joker’s bullet hole tattoos match up with puncture marks on the graffitied Robin costume seen hanging in the Batcave in Batman v Superman: Dawn of Justice.*
But that rumor was just fan fiction, nothing more then that and even David Ayer debunks it in a interview:
Recently, Ayer was asked in a red carpet interview by WSVN-TV reporter Chris Van Vliet to name the most ridiculous rumour he had heard about the film. “That the Joker is Jason Todd,” he said. “It ain’t the case. He’s not.” www.independent.co.uk/
Even Zack Snyder debunked it by mentioning there is a dead robin in DCEU:
"In my mind, it was that Robin had died 10 years earlier, during some run in with a young Joker," he stated. "To me, it was a fun backstory… to play with. The whole idea was that there had been loss and sacrifice. [Batman] doesn’t really have a life outside of the cave. I thought by including Robin – a dead Robin – it would help us understand he’s been on quite a little journey."
Zack Snyder's interview was not that clear. But there was a self proof in Suicide Squad film itself with the intro of Harley Quinn:
Borrowed idea of image from KutuluMike's answer form related question, reading it might clear it more:
No, Joker was not a former Robin. Rather, Joker was responsible for murdering the former Robin, which we saw hints of in Dawn of Justice and then got confirmed in Suicide Squad. (Harley Quinn was arrested as Joker's accomplice to that murder.)
The characterization of Joker in that movie is different from most of the others we have seen on-screen, but Joker's character has varied quite a lot in the comics over the decades. This version of the Joker is much more low-key than others (almost polar opposite of the campy 60's Batman), and is playing more of a mob boss than a crazy terrorist, but he's still definitely playing the Joker.