A 3-1 loss on Saturday ensured United’s worst league start in 29 years.
Jose Mourinho’s side are in 10th place, nine points behind leaders Manchester City, after losing three of their opening seven Premier League games.
“It was not good enough from a Manchester United team with all the talent we have,” said Shaw, 23.
“We didn’t look like a team that was going to beat West Ham.
“If you want the truth I think it was honestly horrendous. I think individually and as a team we were awful.”
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The result at London Stadium followed a home defeat on penalties in the Carabao Cup by Championship side Derby, and increased the speculation surrounding Mourinho’s future.
Earlier in the week, the United boss confirmed midfielder Paul Pogba would not captain the team again, reportedly because of concerns about his attitude.
The France World Cup winner, 25, had criticised the approach during the 1-1 home draw with Wolves on 22 September.
United’s 10 points from seven games is the same as in 2013-14 under David Moyes, but an inferior goal difference means this is their worst top-flight start since they picked up seven points in 1989-90.
Shaw, who looked set to leave Old Trafford in the summer after being criticised by Mourinho, added: “We have to look at ourselves as players. We’re the ones who go on the pitch. The manager isn’t on the pitch, is he?
“Games like this [at West Ham] we should be dominating, creating chances, showing people why we deserve to play for Manchester United.
“It was very poor from us. Very, very poor.”
