I don't have a link to the actual academic article on it at the moment, but there are a variety of ways in which Picketty's science has been examined and found to be kind of bunk.
First, he made a lot of methodological errors, like using tax data as a proxy for income and then not adjusting for a change in tax treatment between one year and the next in the 1980s.
Second, more fundamentally, he's computing inequality without accounting for taxes and transfer payments, i.e. the things that we do to ameliorate inequality, e.g. food stamps, the EITC, and the entire Social Security, Medicare, Medicaid complex. I seem to recall that employer-paid benefits which are excluded from tax, like health insurance benefits, are also excluded.
At the high-level, in terms of big theoretical flaws, he posits this broad "when rate of return on capital exceeds the growth of the economy the rich just get richer" theory which is fine until the rich person dies, gives most of his assets away to charity, loses half of what's left to the death tax, and has more than one heir. People sometimes joke about how economists make a bunch of ridiculous assumptions in their models, and "people live for ever" is actually one of them, and it actually makes sense most of the time (because it usually doesn't affect what you're modelling), but this is the case where it affects what you're modelling.
This all started in 2014, so some of it may have been improved or corrected since then, but Picketty's popularity is still independent of any actual measure of facts or truths in the world. It's related to confirmation bias of the listener. This is why he is used in politics so much. (This isn't special or specially bad; it's why anything is, really. But see it for what it is and look to science rather than politicians for a fight on the facts.)