The Japan international, who arrived at Old Trafford from Borussia Dortmund for a reported £17million in July, has struggled to produce the form that turned heads in the Bundesliga.
Injury has frustrated the 23-year-old's progress, with a knee injury suffered in October ruling him out for almost 10 weeks.
Kagawa, who was substituted early in the second half of United's 1-1 draw at Tottenham on Sunday, knows he has to improve.
"I'm really feeling that I need to perform better," he told the Daily Mirror. "I'm not satisfied with what I have done here at all.
"It's a big and really famous club in the world and I need to overcome lots of pressures and be mentally stronger.
"Unfortunately I got injured, but I have to be better and grow as a player and a person, in terms of contributing to the team's wins and succeeding here."
Kagawa has been impressed by United's ability to turn games around if they fall behind, a distinctive trait of Sir Alex Ferguson's side, and feels he must learn to emulate his team-mates' mental strength.
"I really feel that we have a strong ability to bounce back when we are behind, so people in this team must be really mentally tough," he added.
"We have a really strong squad with many world class players, so it's lucky for me to have opportunity to train with them and play games with them.
"Sir Alex is one of the best managers in the world. What I am really impressed are the numbers of great players he has helped to grow, the numbers of titles he has achieved for the team almost every year, and his ability to mould top players into a team.
"Also, the atmosphere supporters make at Old Trafford helps the team to fight back. As I said, my team-mates are so experienced and mentally strong as I said.
"I'm really lucky to play with world-class players and experience those things, including the history of Manchester United."