I think archaic and outdated imply that it is no longer spoken, or is obsolete whereas this obviously isn't. It's a form of non-standard language like a Creole (which doesn't fit here however). I've been mulling this over and have finally managed to think of the proper word to describe this.

https://dictionary.cambridge.org/dictionary/english/patois

the form of a language spoken by people in a particular area that is different from the standard language of the country:
the local patois

https://www.lexico.com/definition/patois
The dialect of a particular region, especially one with low status in relation to the standard language of the country.

‘the nurse talked to me in a patois that even Italians would have had difficulty in understanding’
1.1 The jargon or informal speech used by a particular social group.
examples ‘the raunchy patois of inner-city kids’

‘We have 1984 today; even if not in the form described by Orwell; since newspeak is replaced by the patois of the gang leaders and international body smugglers.’

‘To emulate (in the specific patois of archivists) is to re-create a work that uses a defunct technology by essentially re-copying it into a current technology.’

Origin
Mid 17th century French, literally ‘rough speech’, perhaps from Old French patoier ‘treat roughly’, from patte ‘paw’.



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Note added at 1 day 22 hrs (2020-05-25 11:55:43 GMT)
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You could also add "arcane" or "abstruse" to emphasise its rarity or obscurity

[she spoke in] an arcane/abstruse Turkish patois

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Note added at 2 days 23 hrs (2020-05-26 12:41:00 GMT) Post-grading
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Glad to have helped (and that I finally managed to remember the right word for this!)