When asked if he agreed with recently-departed Leicester manager Craig Shakespeare that "every manager is only ever four games away from a crisis", Koeman said: "Maybe I am in the crisis.
"Everyone knows the manager's job is a really difficult job because things change really fast.
"Obviously we're disappointed not to go out and win games for the fans. For one reason or another it's not happened but all we can do is keep working hard and we're desperate to get it right.
"If we can control games like we did against Brighton then we'll give ourselves more chance to get it right."
Koeman also confirmed Everton owner Farhad Moshiri and chairman Bill Kenwright had recently visited the club's Finch Farm training ground and is confident he retains their support.
"Yes he was here with Bill Kenwright and more people from the board because they hadn't seen the new building," he said.
"You speak about football of course. There wasn't really a message. They are behind the team and the manager. In football that's a nice thing but in football it's all about results.
"Until now it's total support from the board, yes."