Championship leader Bagnaia had his leg run over by KTM's Brad Binder in a nasty accident on the opening lap of last week's Catalan Grand Prix, having highsided out of Turn 2.
Bagnaia escaped without serious injury and has been cleared to race in this weekend's San Marino GP, despite suffering extensive bruising to his leg.
The Ducati rider's crash was replayed 19 times before his true condition was known on the world feed, with the only info prior to his medical centre visit being that he was conscious at the scene of the incident.
The world feed also showed live shots of his worried family and team waiting for him at the medical centre.
Espargaro, who had his own serious crash in Portugal in March that left him with multiple fractures and out of action until the British GP, says the way incidents are shown "needs to change radically".
"That is not correct, and I'm really pissed off with that and it's one of the topics I want to talk about in the safety commission because I have lived it," Espargaro told Autosport at the San Marino GP on Thursday.
"OK, to show the accident is bad, especially when we didn't know how Pecco was.
"But to show the family… I don't show my family on Instagram or social media. I hate it, because I try to protect them from this [world].
"I am the one who has to take all the criticism and all the injuries or whatever and take all this. But I do not accept that my family is shown on TV and that people out there can even see them.
Source: Autosport