Starting from scratch (again), in 20 minutes of quick searching here's some
sources for you. I'm documenting this just as much for my own benefit as
yours. Usually I feel it's pointless to get too quantitative on HN as it's not the proper forum, and anyone seriously interested could find out for themselves, but since you accused me of lying, and in case someone else accuses me of lying in the future...

Total new prison admittances (i.e. excluding parole violations) per 100000 in
2019 (pre-pandemic). Sources: https://www.gov.uk/government/statistics/offender-management...
and https://bjs.ojp.gov/content/pub/pdf/p20st.pdf (TABLE 8).

* England & Wales: 121.42 <- (72172 / (56286961 + 3152879)) * 100000)

* United States: 120.67 <- (395827 / 328000000) * 100000
* Texas (top 10 state): 166.05 <- (48156 / 29000000) * 100000
* California: 75.35 <- (29764 / 39500000) * 100000

Incarceration rates per 1000000 in 2018/2020 (couldn't find 2019). This is basically percentage of population incarcerated (either at any one moment or at any time during the reporting year, depending on source and details), the metric almost universally used in public discourse. Sources: https://www.prisonpolicy.org/global/2021.html, https://www.prisonpolicy.org/global/2018.html

* England & Wales: 141/130

* United States: 698/664
* Texas: 891/840
* California: 581/549

Note the above admittance numbers do not include pretrial detainment AFAIU; excessive pretrial detainment is a very real problem in the U.S., at least. The U.S. numbers mostly exclude sentences served in local jails (i.e. less than 12
months), but I think do include less-than-one-year equivalent sentences in
England & Wales. However, If you look at the pre-trial graph for Texas at
https://www.prisonpolicy.org/profiles/TX.html, which incidentally also
includes local jail (not prison) post-conviction confinement numbers, it could only hypothetically double the admittance numbers if we assume worst case. Not nothing, but still a far cry from the huge incarceration rate metric gap.

You can find additional numbers for various European countries at
http://www.elpueblodigital.es/wp-content/uploads/2020/04/SPA...
(admittance tables from page 95) and
https://wp.unil.ch/space/files/2021/04/210330_FinalReport_SP...
(admittance tables from page 97). Interestingly, the England & Wales numbers
therein seem to double count pre-conviction and post-conviction
admissions of the same prisoner for the same offense, which seems to be how
they arrived at an almost double total admittance rate of 215/100000 in 2019,
which likely accounts for England & Wales being in the top 25% of EU
countries by admittance rate (see elsewhere in that report). That seems
problematic--perhaps related to definitional treatment of common law vs
civil law detainment procedures?--or maybe indicative of other comparison issues.

There's also the issue of parole. In the U.S., "33% of all prison admissions
nationwide are due to parole violations."
(https://californiainnocenceproject.org/issues-we-face/recidi...)
That's why total prison admittances in the U.S. are 175/100000 (see BJS
source). But in England & Wales, the share of readmissions is ~20% (see
SPACE report), so not dramatically different. That also might be a relevant
source of inconsistency across jurisdictions generally, especially
considering that it seems most EU jurisdictions don't have an equivalent
concept of parole if the SPACE table is taken literally. But ultimately it
can't effect my point as it couldn't remotely account for the sheer
magnitude difference in incarceration rate in the US, and fundamentally is a
matter of sentence severity--longer incarceration durations creating a larger total incarcerated population, which is precisely my point. (There's a scholarly argument--preexisting, not my own--regarding common
law countries adopting more severe sentences to compensate for more procedurally rigorous criminal law protections, whereas punishment in civil law jurisdictions tends to be swift and short, but there's no reason to go
there.)