Leaders Liverpool maintained their seven-point lead at the top of the table even though their match at Everton was postponed because of storms on Merseyside.
Amorim replaced the sacked Erik ten Hag at Old Trafford last month after a terrible start to the season.
He began brightly, with a draw and two wins in his first three games.
But reality is starting to bite hard after a 2-0 Premier League defeat at Arsenal in midweek followed by Saturday's chastening 3-2 loss to high-flying Forest, who capitalised on United's defensive blunders.
United forward Rasmus Hojlund cancelled out Nikola Milenkovic's early goal at Old Trafford, but Morgan Gibbs-White and Chris Wood put the visitors in charge. Forest kept the home side at bay after Bruno Fernandes pulled one back.
United's Andre Onana was embarrassed by a swerving Gibbs-White strike for Forest's second while the goalkeeper, Lisandro Martinez and Matthijs De Ligt all failed to deal with a looping Wood header for the third.
The win lifts Forest to fifth in the Premier League table. United are languishing in 13th spot.
"Tough game in a tough moment but we have to continue the job," a disappointed Amorim told the BBC.
"This is a long journey. We are improving in some aspects. We have to win games to help us improve the team.
"We are in the start of something so big that we pay attention to every detail. We already know it's a big job so we have to continue."
Forest boss Nuno Espirito Santo said "I'm very proud of the way we work together, build things together, from the owner to everyone else in the club.
"It's a moment to celebrate. But for me, I'm thinking about (next week's match against) Aston Villa."
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