I would go with something no higher than a 980/1060/1660 or RX 580 with your 3570k.
You will be massively bottlenecked with something higher like a 1080 / 2060.
Source: Me, a guy who put a GTX 1080 in a system with an i7-3820 and a 1080p 120 Hz monitor.
You could probably score a deal on a used 1060 or RX 580 on craigslist, etc, but be careful for cards from miners. Ask how many the seller has for sale. If they have several, it's a red flag.
i7-8700k, Custom Water Loop | ASRock Fatal1ty Gaming K6 | 16GB DDR4 3200 CL16
RTX 2080 | BenQ 24" 120Hz | 2x 840 250GB | MX500 1TB | 2x2TB HDD | Win 10 Pro x64
X-Fi Titanium Fatal1ty Pro | Sennheiser HD555 | Seasonic SSR-850FX | Fractal Arc Midi R2