Liverpool twice took the lead in Turkey but were unable to protect their advantage, suffering their second consecutive defeat under Andoni Iraola.
Alexander Isak put the visitors ahead in the 69th minute, only for Fenerbahce to respond through three goals from dead-ball situations. Mason Greenwood, Archie Brown and Vedat Muriqi found the net as the Turkish side completed their comeback.
The result continued a worrying pattern for Liverpool, who have conceded at least three goals in each of their last three major European matches, having also shipped three against LASK and Villarreal.
Van Dijk admitted that the defensive performances were becoming a serious concern.
"It's definitely not sustainable," the Liverpool captain said.
"That's why it's a very disappointing night for many reasons, but definitely because of the goals we conceded and the manner we conceded them as well. There is work to do."
The Dutchman was particularly frustrated by the manner in which his side defended the set-pieces, having previously been reliable in that area.
"Three set-pieces as well. We have been very good at them so far this season, but we switched off and weren't ready for the first and second balls. It's a tough one to take."
Asked whether the defeat had angered him, Van Dijk made clear that the players were fully aware of their shortcomings.
"Yes, but I didn't show it because everyone looked disappointed and they have a reason to be – they should be disappointed.
"We cannot lose in the manner we did. But we did and it is down to Fenerbahce as well."
Liverpool had won five consecutive matches before losing to Arsenal and Fenerbahce, and Van Dijk believes the difficult spell must become an opportunity for the squad to strengthen its unity.
"I mentioned after the last game [against Arsenal] that we as a group have to go through good moments but also lesser good moments," he said.
"That will only form a bigger bond and it will be only beneficial. No-one wants to lose games, especially how we did, but it should only help us in the near future.
"We should not accept losing, especially in the manner we did."
Liverpool must now turn their attention to Sunday's Premier League trip to Crystal Palace, with Van Dijk aware that another demanding challenge awaits.
"It's massive. When we are back in Liverpool, preparing and recovering physically and mentally, we will be looking forward," he said.
"It is always tough at Palace and I don't expect anything different."