After Real defender Raul Asencio was sent off on seven minutes, Bellingham soon opened the scoring for Real as they eventually won 3-1.
He later said, "It was a good win for the team, we played very well. We improved on a lot of things from the other day. We're very happy, but now we want more.
"The truth is that the game got difficult after Asencio was sent off. He's a young defender, so it's normal, it's no big deal. It's important to change the system when the coach tells you to, and we had to take responsibility for that. We had to communicate with each other to make it easier for the team.
"Then we had to weather the storm a bit. They had several shots on goal and (Thibaut) Courtois was magnificent. We won. It wasn't a perfect game, but we can keep working. The coach's ideas are fantastic and we have to keep working in the direction he wants."
On the early impact of coach Xabi Alonso, Bellingham added: "I talk to him a lot, every day. He's a very good coach. He has very good ideas about how to play, in defence and in attack. He's really important for us and for this new chapter. I'm very happy with him."