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 which 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 Motorsport.com 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.
"One of the things my wife struggled with a lot was she was on the camera just in front of the hospital waiting for me.
"In the case of Pecco, they told them that Pecco was OK. But in my case I was f****d, I was bad – really, really bad.
"So, my wife was already suffering a lot, plus to be on the TV and cameras there, it's super bad. It's something that needs to change radically and they need to change so much."
Yamaha rider Fabio Quartararo called the constant replays of the crash "unacceptable", especially with the riders having to go back out and race afterwards.
"For me, it's unacceptable," the 2021 world champion added when asked by Motorsport.com about crash replays.
"You can show it once, but in the end, all the other guys are starting after. You put it one time on replay, but you need to think there are 20 other riders who need to restart and it can happen to you.
"So, mentally it's difficult and you see one big crash. In the end, the rider is fine but it can be much worse.
"We know when they put 'rider OK', you don't know if they've broken one arm and two legs, or nothing. So, this for me, to have it one time is OK.
"But to have it on repeat before our race is not nice."
Marc Marquez agreed that it's difficult for the riders, and feels 19 times for the replay of the crash was excessive.
But he also notes that it is simply part of the normal broadcast process and crashes generate more engagement than any other type of content.
"It's true that for the riders it's very difficult to see that crash many, many times, especially when you need to go out again," the Honda rider added.
"Pecco was good and like this they said he was good. Dorna had the information from the ambulance that he was good, just they needed to check everything [was not broken for Bagnaia].
"It's part of the show. It's true that it's not necessary to show many, many times maybe. But it's not my decision.
"In my case, I turned off the TV and I was fully concentrated on my job. But it's true that if you check the views on the internet of the crash and a race victory, there's more views for the crash.
"If the people want to see that, the guys who produce the image must show. But it's true for the riders it's very difficult. When I re-checked the race at home, I passed that moment because I didn't want to see it as a rider.
"If it's 19, then it's enough. Five is enough, or even less. If it's 19, I will say that's too much."
