The Italian claimed his side had been "almost perfect" during the first half of their Champions League trip to the Santiago Bernabeu, but says defensive errors cost them dearly
The Turkish champions had gone in all-square at half-time, with Umut Bulut having cancelled out Gareth Bale's free kick, but a second-half collapse saw los Blancos power home to an ultimately emphatic 4-1 win.
The result was made all the more exasperating for Mancini by the fact that Madrid defender Sergio Ramos had been dismissed midway through the opening 45 minutes.
"We were nearly perfect in the first half," the 49-year-old Italian told reporters after the game in the Spanish capital.
"I really don't know what happened in the second half; I need to talk to my players."
"If you make mistakes against a side like Real Madrid then they will punish you. They have very strong and talented players."
"We have defensive problems, we need to work harder because we can't win with big mistakes like these."
However, despite dropping in thrid position in Group B, two points behind Juventus, Galatasaray can still advance to the knockout stages with a home win over the Bianconeri on the final matchday.
"One point was not enough for us because Juventus won," he said, in reference to the Italian champions' 3-1 victory over Copenhagen in Turin.
"Now we will prepare for that match. It will be decisive and we need big support from our fans in that game."