Proposing IRV is nice and all, and definitely an improvement (whatever you do don't listen to nutters proposing range voting…it's trivially gameable…and personally I just don't think Approval's lack of ability to give a nuanced vote is very good).

But the real change happens when you move away from single-winner seat entirely. Use something like MMP or STV where the votes can be distributed proportionally.

Australia is a really good example, because we have a bit of both. Look at our House of Representatives. It uses IRV like you propose America switch to. Labor got 33% of the vote and 51% of seats. LNP got 35% of votes and 38% of votes. Greens got 12% of the vote and less than 3% of seats. Yikes. One Nation got almost 5% of votes and 0 seats.

Then look at our Senate. It uses STV so that in a normal election, each state elects 6 Senators and territories elect 1. Labor got 30% of votes and 20% of seats. LNP got 33% and 20%. Greens got 13% and 6%. One Nation, United Australia Party, independent David Pocock, and the Lambie Network each got 1 seat (1.3% of seats) on 4.3%, 3.5%, 0.4%, and 0.2% of votes, respectively. These numbers are obviously not perfect, but they're a hell of a lot better than the Reps' results. STV is a sort of quasi-proportional system, retaining local representation. In our case "local" means "at the state level", but you could also do it by taking 5–12 House of Reps districts and merging them into 1 district, returning 5–12 Representatives.

True proportional systems like MMP (look at NZ and Germany for examples of that in action) or direct proportional systems without a local member (like the Netherlands) get even closer to perfectly matching voters' will.