The problems Pellegrini faces in terms of criticism from City supporters - and the criticism various players get - has been done to death. The guy has never been able to win and, as I have stated before, that is one very big reason why I try to focus on his positives rather than his negatives. Yesterday, we saw his faults exploited to a level that in fact does not happen often. His modus operandi is to try and stick to a way of playing and he does do it for a reason, which you can like or dislike; however, City have consistently struggled against sides that press really hard (it was also the case under Mancini) and with the current squad it is hard to excuse him not picking a different starting line-up and approach for this game, so I won't.
That said, I do think that his players have performed poorly in ways that are not excused by the manager's team selection or tactics so you could say they let him down, again. The first City goal, which the team never recovered from, contained two crucial individual errors that cannot be blamed on the manager.
I suspect that yesterday may have delivered a mortal blow to Pellegrini if the plan is not already to replace him at the end of the season. I just hope that the club is not found to have made a big mistake in not going for Klopp when he was free in the summer. In the meantime, City still have everything to play for and Pellegrini needs to ensure that his team do not lose two games or two league games in a row. The priority in Turin needs to be a clean sheet and he needs to pick a side that is fit for purpose.