The distribution of the tax burden is obviously a politically loaded question, where those on the right tend to have an interest in claiming that the US tax system is already (too) progressive, while those on the left wanting to increase the share of the burden shouldered by the rich. However, the question of how progressive the system is an empirical question. In fact, the system of taxation in the United States is relatively progressive. What makes this fact surprising is that tax progressivity and fiscal redistribution (the reduction of inequality by government action) are often conflated, and it remains true that redistribution in the US is low, due mainly to the relatively small size of the US government. The American case typifies one pole of a robust negative relationship between tax progressivity and overall redistribution, the cause of which remains a contested question in the literature on comparative political economy.
I’m hesitating a bit to post this, because there are tons of ways that claims and studies like this can be biased, but I think the article is thought-provoking and worth reading. If you end up finding responses by respectable economists, please comment.