If I could reach into peoples' heads, there's a solution: "mind your own damn business". If we didn't care so much what other people did, then we wouldn't disagree so much. And in general, I think the world would be a better place if more people did this. But some fraction of issues will always affect multiple people (that's the whole point of government, after all).
And besides, nobody's yet invented a mind-control ray, and it would be pretty icky if they did. (There's a certain irony to altering peoples thoughts so that they care less about what other people think...)
For instance, any decision that will last for generations and impact profoundly in everyone's life, like secession or a "Brexit" should require slightly more votes than a majority one. Unfortunately it's only typically used by parliaments, courts and senates for constitutional change, never for popular vote on issues.
Simple majority has the advantage that it's fair. If you're going to start claiming that decisions that will "profoundly impact everyone" need higher than majority you're just going down the road to totalitarianism. Anyone can claim a decision they think they're on the losing side of is too important to leave to majority decision making. Then it's just about ramping up the threshold until you win.
(I think) In democracy, voters are less likely to compromise. This is result of weak discussion between >1m.
But likely technology (as always) will come to rescue before democracy. This is in reference to rise of renunciations, AirBnB etc.
Supermajority voting on all legislation & referenda? After all, 2/3 of the population forcing 1/3 to obey a law is much more reasonable than 51% forcing 49%.
And making amendments take a 3/4 vote is even better: if 75% of people think something is a good idea, then it probably is.
Secondly, for spectrum decisions (e.g. what tax rate we should have, how much to spend on defense and education, etc) getting as close to 51v49 for the decision is probably the most equitable outcome.
America (and the few other remaining non-proportional systems) suffers from this problem much more than the more advanced democratic systems. If America is all you know, you shouldn't generalize too much.