How to fix very high impact of losing players last turn
While I haven't played Game of Thrones, kingmaking is something that happens in a ton of games.  And I've only ever found a couple ways of effectively dealing with it.


Embrace it.  Sometimes, it is just part of the game.  Sometimes picking an otherwise sub-optimal move in hopes of currying favor for a possible (or probable) kingmaker scenario is a valid tactic.  Not snowballing someone to oblivion can gain enough favor that you win the table politics.  If you choose to embrace it, realize that table politics can become far more important in your games.  It can also make those games more interesting (or more frustrating) depending on your feelings towards table politics.
Treat it like luck.  This may not work if kingmaking scenarios happen often, but you can treat it like any other luck-based or random event.  Just like when you need to draw that certain card or get that high roll to snatch victory from the jaws of defeat, sometimes you just have to hope a kingmaker chooses you over the others.  Winning or losing because of a kingmaker can feel like a cheap win or loss.  It can also be a learning opportunity into how you should play to avoid such scenarios if it bothers you.  Also, like other random events, sometimes you just get screwed.  Don't let it bother you and move on.
Encourage not being a jerk.  If this really bugs you or people in your playgroup, encourage people who become kingmakers to just stay out of it, especially if their motives are less than sportsmanlike.  Just like with any other not-technically-illegal-by-the-rules-but-still-a-jerk move, tell people you don't like it when that happens.  If one person keeps doing it, stop playing with them (and possibly politely explain why).  You play the game to have fun.  Don't play with people who suck the fun out of it.  Either they will learn and stop or they won't be around to suck out the fun.1


1 I realize this is an easy thing to say and a much harder thing to do without bruising egos and hurting friendships, but setting healthy boundaries is essential to good relationships.  And if someone can't respect those boundaries (or at least agree that maybe playing a certain game isn't good if they value you as a friend) then they might not have been a very good friend to begin with.