The 25-year-old came into the race aiming for the Tour's top spots after years of improving, encouraged by twin brother Simon's ride in the Giro d'Italia this May, but now finds himself 16th overall, 5-51 behind the yellow jersey.
"We knew it would be an aggressive day," Mitchelton-Scott sports director Matt White said after the stage.
"We put Mikel [Nieve] and Damien [Howson] in the break. We called Damien back from the break to be with Adam. He didn't have the best of days there in the second group. He was a little bit isolated in the second climb."
In 2016, Yates placed fourth overall behind winner Chris Froome and won the youth classification, but now faces an uphill battle if he is to come anywhere close to that result after he suffered from the heat on the race's first summit finish.
"I haven't spoken to Yatesey, but he went from being very good to exploding in a short period on that final climb," White continued.
"He is not a fan of when they [Sky] really accelerate, so at first thought he was just riding his own tempo. Then he was asking for water – he was already inside the last 10 kilometres – so he was obviously affected by the heat."
Watch: Tour de France 2018 stage 11 highlights
Yates faces another hot day in the mountains on Thursday with stage 12 crossing two hors-categorie climbs on the way to Alpe d'Huez.
"He can only recover as fast as he can recover. We have seen other guys in the last few days have bad days. It can catch up with you and the next day you can be back in the front group again. We just have to stay positive," said White.
"It all depends on how much he loses tomorrow and if he is better or worse. We have to reset, recover and get through a tough day tomorrow and then we will have a much better idea of what we can chase after tomorrow's stage."
The Tour de France travels west before more mountain days in the Pyrenees and the final time trial stage. The Tour concludes in Paris on July 29.
"The GC guys have three days to relatively recover – Friday, Saturday and Sunday – before they hit it again next week," White said.
"First thing first is to recover and see what we can do tomorrow and then we will have a much better idea of whether we give up on GC and chase stages or continue in GC mould and obviously it is going to be good for his development as well – he is only 25 – and whether we are chasing a top 10 or not. We will see what we are doing in 24 hours time."
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