I'm a US citizen living and working abroad, married in 2016 to Romanian woman. Filing separately. I can't proceed without entering a SSN for her but she doesn't have one!

If you are a US citizen and your spouse does not have a Social Security number or an ITIN and you are not applying for an ITIN with the tax return then you can only file your tax return as Married Filing Separately.  Where asked to enter the spouse's Social Security number enter 999-88-9999.  You can only print and mail your tax return, it cannot be e-filed.  When you print the tax return erase the Social Security number for your spouse and manually enter NRA for non-resident alien.

See this TurboTax support FAQ for the procedure to print and mail a tax return using the online editions - https://ttlc.intuit.com/questions/1944348-how-do-i-print-and-mail-my-return-in-turbotax-online

Your spouse would only need to report their income if you were filing your tax return as Married Filing Jointly, applying for an ITIN with the tax return and you would be providing a statement with your tax return that you want your Nonresident Alien Spouse Treated as a Resident.  See IRS Publication 54 Tax Guide for U.S. Citizens and Resident Aliens Abroad pages 6 and 7 - https://www.irs.gov/pub/irs-pdf/p54.pdf

Go to this IRS website for ITIN information - https://www.irs.gov/individuals/international-taxpayers/general-itin-information

If you do not have a US phone number for a callback you can contact TurboTax support using these websites -

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