I went with the institute in fallout 4 and everyone’s making me feel like shit about it. Nobody in the commenwealth trusts the institute, and are very weary about synths roaming around. Is there a way to redeem the institutes reputation and have everyone trust them? I haven’t done any missions yet as the director or the institute, But I don’t want to do missions for them if it means I’m going against what everyone else in the commonwealth seems to want. Also by taking out the railroad and the brotherhood I feel like I cut the story in half.
No, unfortunately you cannot change the general opinion on factions.
A pity, because the Institution would make a hell of an asset to any other organization, not to mention to the common ..wealth.
The main reason for the impossibility of reconciliation is a conflict of interest between almost all of the factions, embodied (in Fallout 4) simplistically and symbolically by the Institute's synths:
the Institute creates synths to enforce 'peace' in the Commonwealth, alienating all other people in the process;
the Brotherhood wants monopoly over all technology, including the synths;
the Railroad wants to free all synths from their puppet masters;
the Minutemen want to protect the common people against any danger.
More on the factions here.
So, siding with any of the three antagonist factions - the Brotherhood of Steel, the Railroad, or the Institute - initially lets that faction be more or less tolerated by the other two, up to the decisive moment in which you take a stance against (either of) the other two.
The Minutemen will always remain neutral towards a player in alliance with any of the other three factions, deeming the player capable of knowing how to achieve reasonable peace.
Here's a similar discussion on the Steam forums.