I don't see Koeman turning this around. Forgetting the transfer window failures, he's just been very slow to react to the problems we've seen on the pitch and he seems to only get certain things right each week, but then persists in making mistakes in other ways.
For example today. He's started Niasse and Calvert-Lewin up front which was quite positive. Started Vlasic to provide some width on the right and left Rooney, Klaassen and Sandro out who were 3 players he brought in over the summer. However there were far more mistakes in his line up than positives. He's kept Williams in ahead of Holgate, gone with Martina instead of Kenny, played Sigurdsson out on the left, played 2 defensive midfielders, dropped Davies, gone for Sandro ahead of Lookman when making changes. These are all persistent things he seems to be getting wrong.
With Williams, he is struggling at the moment and part of it is down to the way the team is set up ahead of him. When Barkley was in the team, the more limited players on the ball - Williams, Gueye, Schneiderlin could do what they do best which is win the ball and give it to someone like Barkley and the expectation would be on him. We have a situation now, and we had it with Jagielka against Stoke, where Williams is uncomfortable on that left side of the 2 but he is trying to force things because no one ahead of him is giving him much of an option. Holgate may have made mistakes but he can't do worse than Williams in my opinion. Martina is another one who is being favoured by Koeman and only he can see why.
I think all of these mistakes will result in him losing his job in a month or 2 because the fans can see the way this is going and we aren't going to pull ourselves out of this mess unless Koeman starts correcting these mistakes. I've seen nothing to suggest that he will do it though. Far too stubborn.