I own a 13 model too and have driven 21k miles with no oil issues at all and change every 3-4k at the dealership with synthetic.
Sounds like you just got a defective engine with yours. This rarely happens but it does.
Sorry to hear you having issues with your Z but I believe most owners here do not have this problem.
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Takeda TR3009P Short Ram Intakes, Custom Dual Exhaust, X-Piped with 18" Resonators & TopSpeed Axle Backs. EVO-R Led Third Foglight,
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