The derby loss was their third straight defeat, but the Argentine has urged fans to keep faith in his abilities as a coach.
"I do not like to boast, but I know what it is and I'm facing [our slump] with the greatest possible firmness," he told reporters after the game.
"I have been in football for 30 years now, so this is not new for me. I've faced these types of situations countless times before.
"I have enough resources at my disposal, but the willingness of the players comes from the coach who in turn needs support from within the club.
"However, accountability is very serious and I cannot feel good at the moment."
Reflecting on the loss, Bielsa admitted that he was beginning to grow tired of his side consistently repeating the same unnecessary mistakes.
"[Sociedad's equaliser] was not a factor in our defeat because we must be prepared for all possible outcomes," he added.
"There was no reason for the disappearance of our superiority we had over our opponents and I have nothing else to say, I've said all of this many times before.
"Our tendency to play well in the first half and fail to see it through in the second has always been our problem and we've not been able to resolve it."