This seems to be a recurring issue that has been reported on other online forums as well, and can happen on any gaming platform. So far the only "solution" is using the console command on PC, which is unfortunately not an option for console players like me

In short, the problem is that after talking to father and siding with the Institute, the "Institutionalized" quest won't finish, so no new quest is triggered. There is no quest mark on my map, and I cannot fast travel out of the institute either. I am essentially stuck there and the game becomes unplayable. One possible explanation is that this happens because Dr. Li did not properly install the courser chip in me (e.g. if you are wearing power armor or diverted the conversation when she mentioned installing the chip to you).

I am a new user and I'm not allowed to post links, so for other threads reddit/steam for this problem, please do a search with the keyword "fallout 4 stuck in institute".

Next, for my case:

I'm playing on ps4. I sided with minuteman before the quest institutionalized. When Father first asked me to join the Institute, I said I am unsure. Then I went to talk to the department heads, and explored the Super Mutant lab, collecting the cure for Virgil while I was at it. I wore power armor the whole time and when Dr. Li asked to install the courser chip in me, I asked her something else. This may have caused the bugged for me.

At some point, the quest log prompted me to "exit the institute", and a quest marker is placed at where I first teleported into the institute. Instead, I went back to talk to Father, said I changed my mind and decided to work with him. After I do this the quest mark just disappeared. The "exit the Institute" sub-quest also disappeared, but the "Institutionalized" quest itself won't terminate. If I talk to Father, he just asks me to talk to the department heads and say how he looks forward to working with me. The department heads won't give me anything new either.

As a side, I made sure I collected the holotape for Sturges, so that shouldn't be the cause for this either.

Devs, please fix this... It's one of the main quests and my last hard save was 1 week ago (now I know how stupid I am for not saving more often...).