America Versus Europe

US growth has been relatively strong the last few years while European growth has been barely positive.

This is a bit too vague for me to answer definitely. The question is about changes, not levels, and medians not averages. Hard! I imagine the answer is Yes for most European countries, but No for a few Eastern European countries (Poland, Romania) that have grown A LOT since 2001.

This is not what the data tell us. It is important not to rely on market exchange rates, but rather to think in PPP terms. In PPP terms, there is no such decline. Furthermore, productivity has increased at a similar rate in the US and the EU.
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Which income, pre or post tax? It is not clear that pre-tax median income in the US has grown more than in Europe with a higher increase in inequality (US mean has grown more than median). And post-tax median income with more redistribution in most European countries even more so

There has been a decrease but I would not use the word substantial given the magnitude.

The income of the mean citizen in Europe has declined relative to the mean citizen in the US. For the median, I would have to look closer into the data. Also, how do we define the median European citizen? Do we treat Europe as one country?

The key is median: income inequality in Europe as a whole seems to have been declining, but not in the US. If the median income gap is widening, probably it is not by much. Garicano points are valid, but would not sufficiently affect this particular metric..

My hunch is that the discrepancies in income between US and EU have risen at the upper and lower quartile in opposite directions - but they were constant for the median household. The gap is negative in the upper quartile (US>EU), and negative in the lowest quartile (EU>US.

It really depends on where the median European citizen lives. It may well be in a country like Czechia or Slovenia, which converged toward the US in the last 25 years.

There is relative decline for median citizen or household but it is modest when income is adj. for taxes, inflation & PPP. It is more pronounced using GDP per capita at market exch rates or when comparing top of income distribution. Also not a decline against all countries.
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Median is different to mean. US median income stagnated post GFC until around 2018 then rose post COVID. Former communist bloc EU countries median income grew considerably during the last 25 years.

This is a fact. It is iInteresting to discuss whether different definitions of income for comparison purposes make the gap more or less substantial. But a fall is still a fall.

Median income growth has been roughly equal in the US and Europe since 2000, although there is large heterogeneity. Income growth has been higher in Eastern Europe and lower in Germany

PPP-adjusted disposable income gap between EU and US is approximately the same today (~35% lower income in EU comp to US) as in the early 2000s. Differs across countries, though. E.g. has increased w Germany, Italy and Greece; decreased w Sweden, Romania, Poland,...
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European median incomes appear to have declined when expressed in USD. But the picture is less clear when doing PPP adjustments. Much more work is needed to agree on this basic indicator.

If measured as per capita GDP PPP adjusted

It depends how the term living standards are interpreted. In terms of incomes the median European is not doing well. But in terms of lifestyle people in countries like Italy and France live well.

Again, lots of heterogeneity there. Plus, it matters whether living standards are just about cost of living, or about other things such as life expectancy or other non-economic factors.

I would even say quite the contrary. First, there is no relative decline in income. Second, working hours are lower in Europe, and on average other indicators, such as health outcomes, are better as well. Overall, European countries therefore seem to perform better.

Substantially lower sounds much stronger than what the data suggests.

Too much lifestyle variation across Europe, and different degrees of government provision of welfare services to be sure. Measured disposable income differences are not dispositive.

The attention of observers is mostly focused on the upper end of income levels and living standards where US level seems to be substantially above the European ones. But for the median household these differences are much smaller, or do not exist at all.

The median European citizen does not live in the highest-income countries.

Large gap if liv stds = GDP/capita at current rates. But adjusting for PPP & hours worked, US & North-Western Europe are close in income. Not true for all of Europe. If liv stds include health care, education, public services (& happiness), no gap or much of Europe ahead of US.
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Various pieces of evidence show much higher life expectancy, lower infant mortality rates and morbidity, and much lower deaths of despair. Regional and ethnic inequalities are much lower in Europe. Even upward intergenerational mobility is HIGHER in many EU countries.

Besides lower incomes, Europeans also have less personal comfort: smaller homes, no air-conditioning, fewer cars, more red tape. They tend to work less and compete less, which can be viewed as a plus or minus depending on how much you like your job and how competitive you are.

Statistics do not lie.

Median disposable income is roughly twice as large in the US. But certain government services such as health care are still better in Europe. On balance, living standards for the median household are better in the US

PPP-adjusted disposable income in EU is clearly lower, but accounting for dree healthcare, free or highly subsidized higher education, and more leisure time, I would say that median quality of living is higher in EU comp to US.

Going beyond real incomes, Europe seems to perform better when it comes to other indicators relevant for wellbeing, including health (e.g., life expectancy), violence (e.g. homicide or overdose death rates), etc.

Less income but better place to live for low incomes.

If measured in real domestic terms and taking into account life quality (leisure, life expectancy, health services, etc.)