In Irish you would write this as tá mo charr mór, but Irish doesn’t distinguish alienable and inalienable possession the way Sc. Gaelic does (in Scottish my car is an càr agam rather than
mo chàr like Irish mo charr).
As kotiteollisuss wrote, tá carr mór agam and tha càr mòr agam would rather mean I have a big car, different type of sentence.
Hypothetically you could, it’s not ungrammatical. But since car is not an inherent part of your body not a family member or a very close friend, it’d rather be referred to using alienable possessive construction: an càr agam rather than
mo chàr in Scottish Gaelic (but then, in Irish it would be mo charr).
See Possessives and syllabic structure or Ar n-Athair a tha air nèamh on the Akerbeltz wiki for more details.
That’s because Duolingo is nitpicky about grammar – you’re sentence basically means the same thing and has similar structure but note that you have another verb: the car I have is big, here I have introduced another verb in a relative clause – and you can do this in Gaelic too.
So for your translation to work in the nitpicky Duolingo exercise, in Gaelic it would have to be tha an càr a tha agam mòr (but then you’d probably rephrase this in Gaelic to something like tha e mòr, an càr a tha agam or an càr a tha agam, tha e mòr, lit. ‘the car I have, it is big’ to avoid the nested tha clause…).
an càr agam isn’t really the car I have because it lacks any verb in Gaelic, it’s literally just the car at/by me and means my car. Maybe the car of mine would be good English approximation – but not sure if that’d be accepted either. Though you certainly can think of it in terms of the car I have.
Hope this makes some sense to you. :)