Brett Pitman profited from two first-half penalties to put City in command and the visitors' problems worsened when Richard Cresswell received a straight red card for fouling Marvin Elliott just before the break.
"Decisions have gone against us this season," said Speed. "But to concede two penalties and have a man sent off is unprofessional and we showed a lack of discipline at times.
"I thought the first penalty was the correct decision. The second was more debatable because their lad has hit a shot from about a yard away from our defender, who was judged to have handled it.
"The sending-off was silly. It may have looked harsh, but Cresswell went in with two feet raised and you cannot do that in the modern game.
"You see tackles like that punished with red cards on television every week. Even at 2-0 down I fancied us to get something from the game, but going down to 10 men made it impossible.
"Irrespective of the decisions, we didn't play well enough. Bristol City were the better team throughout and when that happens you have to roll up your sleeves and really battle."
Robins boss Keith Millen has seen his side make a steady improvement following a nightmare start to the season.
He said: "That's one defeat in eight games for us now and we are certainly heading in the right direction.
"We don't like looking too far ahead. We take games in groups of five and set ourselves a points target, which seems to be working well.
"I thought our wingers caused United problems all afternoon and created the space for our strikers to work well together.
"It meant we were able to really stretch our opponents when they went down to 10 men and I was pleased to see us dominate the second half.
"We've created a bit of breathing space between ourselves and the bottom of the table, but there is a long way to go yet and a great deal more hard work ahead."