How to manage the relations between supervisor and institution? For a student, the eventual conflict resolution should be facing the supervisor. This already generates a first problem: how can a PhD student explicitly face uncomfortable situations that imply a lack of seriousness and correctness of your supervisor, since you completely depend on him in the present and for the future?

A second problem can be that the student might not trust how the supervisor suggests to overcome some issue or how to develop further the progress of the PhD (not the research itself).

Supposedly, it might be possible to bypass the supervisor by attempting to get some suggestions or information (from the institution) that clarify that the horizon of possibilities is not limited to the options given by the supervisor. Indeed, the supervisor might have designed your professional development (which you don't know and you have never agreed) by creating constraints that do not necessarily exist. Then, it can happen that the institution is forced to necessarily defend the way the supervisor operates, because differently would mean asserting that there is something wrong. Thus, is easy to expect that the institution is going to push you back to the hands of your toxic supervisor.

I'm experiencing this condition as a tennis ball that has to find a way to accommodate the supervisor and institution, without forgetting that I might even want to decide what to do, by having clear the large picture and taking my choices consciously. This is a right that both institution and supervisor have purposely neglected to provide (making me fully dependent on them).

Currently the situation (which is flipped in comparison to my initial story) is that the supervisor wants to clean up some mess he made by helping me to get to the end of the PhD (whatever it is). On the other hand, now the institution seems to be concerned about its correctness. I end up being between the hammer and the anvil and I don't know if I should expect to be defended by one of them or I should push one against the other to solve their own problems, in both cases I don't see a way out.
You don't say what country you are in, and what instutional checks and balances exist is going to dependent on that. 

In the US, you will have a thesis committee that is (at least supposedly) indepedent of your supervisor, and, at least in theory, pulls rank on your supervisor when it comes to matters of your studies. In the UK there often isn't a committee, but almost all universities have a system of "second supervisors" or "academic advisors".

In both cases there will be almost always be a head of graduate school, or directory of graduate studies for the department, school or faculty, as well as a chair of the department/school/faulty. The head of studies nomianally out-ranks the supervisor, although politics can come into play here, but the chair of the department/head of school/vice-chancellor of the faculty genuinely does outrank the supervisor. 

Failure to progress (i.e. not doing well enough to continue your studies) is a disciplinary matter, and would be dealt with in a quasi-judical manner according the instututions rules and regulation. You will be able to find these online. 

In the UK there will be Student's Union, and perhaps even a Graduate Students Union who can represent you if you feel you have not been fairly treated by either your supervisor, your department or the university. I don't know how many US institutions have similar. In the UK you can ultimate complain to the government ombusdsman, the office for students.