For the life of me, I can't understand...how do I owe $5k US?

My taxes and tax scenarios have always been pretty straight forward in the US and I thought they'd only be mildly muddled here.  I'm just an employee, getting PAYE deductions and W2 Single/0 deductions.
Here is my 2019 tax year and simplified numbers:

1/1/2019 - 3/10/2019:  US resident, US employee.
W2 - $33k Gross Wages,  $4,975 Fed income taxes withheld.

3/15/2019 - 12/31/2019: UK resident, UK employee
Total wages fx to USD:  $66k Gross,  $15,975 tax,  $4,600 NI (lumped as tax)

Don't qualify foreign income exemptions so I used Foreign Tax Credit in TurboTax, which says I owe $5k to IRS.   

I've never paid quite so much tax and fees as a % of income before so can someone please tell me what I did wrong?

Do you have your residency in there correctly?

https://www.irs.gov/individuals/international-taxpayers/foreign-earned-income-exclusion-physical-presence-test 

Sorry, edited to add - you should qualify for FEIE if that's the route you want to take. Your income in the US and UK is less than $105,900 from the numbers you listed. You'd probably want to compare the results for credit vs deduction if everything is input correctly. 

I think you should be able to qualify for the FEIE.

If you would like any assistance, feel free to send me a message.

Electing to claim the foreign earned income exclusion is frequently a very bad choice, it may not produce the optimum result.

I'd suggest running the numbers with and without the foreign earned income exclusion to see which produces the better answer.  The software you re using may be the problem as it is not designed well with international tax in mind.