I think you are confusing different forms of taxation.
If you are talking about income taxes, then yes, on average, the poor may pay little, however that can also be debated.
There are also consumption taxes. Sales tax, or value added are examples. Poor have no choice but to pay them, and for rich they are peanuts (example of regressive taxation).
Suppose there are 2 people with incomes 100 (poor), rich (10000).
poor man buys essentials and assuming sales tax is 15%, he can buy 85$ worth (for argument's sake as actual amount is slightly higher).
rich man, buys the same essentials, and pays 15$ in taxes and has disposable income of 9900.
for the rich man, he pay 0.15% based on his income, while poor man pays 15%.
Poor man probably works for someone, hence pays income taxes too. Suppose on average 20%. Hence his real disposable income is 65$ not 85.
Rich man, probably has smart dudes working for him to minimize taxes. His corporations probably give him dividends (15%) or tax havens or double irish dutch sandwich (2-3%).
The way i see it, poor man does get screwed.
Poor may pay little in absolute terms, but compared to what they earn and need to stay alive, it is a significantly higher percentage.
Also, poor probably have to go about things more inefficiently than the rich. Poor have to deal with pay day loans (hefty %), transport (spend 2-3 hours on the bus to commute/shop) whereby a car would be faster. ... and so on.