It's not just Twitter, and it's not just "politics." Social media, as currently configured, promotes the wide dissemination of rumor, false information, misinformation, propaganda.
If their use of social media is a fair indication, an awful lot of people are extremely gullible, or just plain stupid - because they routinely "share" information that is undocumented, inaccurate, misleading, or downright nonsense... and treat it as if it were meaningful.
Are we really that stupid, as a society? Is this level of idiocy the reason the Founders designed a representative democracy - to keep the fools from dominating conversations about politics (and policy, and science, and health, and pretty much everything) with ignorant, uninformed myths and lies?
I don't know how we fix it...short of eliminating the ability of anybody on social media (Facebook, Twitter, others) to "share" anybody else's contributions. No "Share" option. No retweets. This wouldn't prevent ignorant people (or malevolent actors) from posting rubbish - but it would greatly reduce the circulation of such rubbish.
Facebook should be dominated by pictures of grandchildren - not propaganda. Twitter should typically be people posting where they got lunch - again, not propaganda.
But these fixes will not happen. Private enterprise, you know. It's going to destroy our political culture, and likely our form of government - because the only way to deal with widespread, widely believed propaganda nonsense, in the absence of a truly educated population (which we do not have and will not have), is to crush it, and crush the people who promulgate it. In other words, expect the rise of authoritarianism, and the loss of freedoms. Expect the kind of things Trump wants to do to undocumented people, but directed at anybody who disagrees with whoever is in power (which will be the wealthy, corporate interests).
I'm 65. I get to watch it deteriorate, and then to die; I don't think I'd want to stay around and watch, if I were a generation younger. (And I may be optimistic here; it's a question of how rapidly things deteriorate.)