The Hoops sit sixth in the cinch Premiership after losing all three away games and Postecoglou's injury concerns mounted further this week.
But the 56-year-old believes his side will emerge strongly from their fitness issues and predicted the cinch Premiership title race would be close this season.
Postecoglou knew he faced a major task when he took over a team that had finished 25 points behind Rangers last season and were in need of a major overhaul of personnel.
Having sold the likes of Ryan Christie, Kristoffer Ajer and Odsonne Edouard and offloaded players such as Olivier Ntcham and Leigh Griffiths, as well as four loan players from last season, Postecoglou eventually managed to rebuild his squad, although not as quickly as he would have liked.
But he could not have foreseen the injury situation which will deprive him of eight players for Sunday's visit of Dundee United, most of them in the forward areas.
New striker Giorgos Giakoumakis suffered a calf injury in the warm-up ahead of Celtic' s 3-0 Premier Sports Cup win over Raith Rovers on Thursday and Mikey Johnston was absent again four days after making his first appearance of the season.
Attackers James Forrest, Kyogo Furuhashi and Karamoko Dembele are still missing along with defenders Greg Taylor, Christopher Jullien and midfielder Callum McGregor.
"With Cal, next Thursday is being optimistic," Postecoglou said. "With Kyogo, I doubt we will see him this side of the international window but he is pushing hard.
"Unfortunately, Mikey Johnston picked up a niggle on Wednesday so he will be out and I'm not sure when he will be back.
"To cut a long story short, we won't get anybody back for Sunday.
"I haven't been through too many periods in my career like this where it's so challenging with what's happening off the field in terms of injuries and disruptions.
"But it is what it is and what our supporters want to know is that we come here on Sunday and do everything to get a result and just get on with it.
"I don't think it's an excuse but there are valid reasons why we are a little bit inconsistent in terms of the way we are playing at the moment, because we have lost our captain, lost our top goalscorer, other guys who can play in the front area haven't been available.
"Those kind of things you can use as reasons for not progressing, but we are not doing that.
"If we haven't performed, we take it on the chin like we did on Sunday and come back in midweek and get the job done.
"If we get through this period it just means we will be a real, strong unit."
Celtic did not have an attacking option on the bench against Raith and ended the match with winger Jota hobbling about in the centre-forward role, although his ankle knock will not keep him out of Sunday's game.
Despite the shortage of first-team attacking options, Postecoglou seems unlikely to go into the B-team ranks to find some cover.
"(Adam) Montgomery is 19, we are playing him here, there and everywhere," he said. "(Liam) Scales made his debut on Thursday night.
"We are either debuting or introducing players every game. We are dipping into the well as much as we possibly can."