The London club were beaten 4-2 at West Bromwich Albion despite dominating for long periods and are fourth from bottom with three points.
Bilic, who said he was embarrassed when they trailed 3-0 at halftime, took his share of the blame for West Ham's woeful start to the campaign.
"We have a problem, starting with me," Bilic told the BBC. "I know that we have a problem and I am the manager and I have to do everything to turn this around, solve that problem."
West Ham have the worst defensive record in the league, having conceded 13 goals including eight in their last two outings.
While they controlled possession at the Hawthorns, Bilic said they deserved nothing from the game having made a number of errors at the back.
"They deserved it. I felt embarrassed at halftime," he added. "We were three down and we can say that some of the goals were individual mistakes.
"We can't make so many mistakes when we don't have the ball. The problem is big, we can't say we were just unlucky."
After conceding a fourth goal early in the second half, West Ham fought back with an effort from Michail Antonio and a Manuel Lanzini penalty.
"There were some positive things in the second half," Bilic said. "But we want action not reaction."
Nacer Chadli scored twice, one a penalty, while Salomon Rondon and James McClean also struck for Albion.
(Reporting by Toby Davis; Editing by Tony Jimenez)
