Old Pacific Highway, AKA 'Old Pac':The good part of the road starts just north of Berowra and it ends around Somersby. It runs adjacent to the F3/M1 freeway, crossing over or under it at various points. On a good day (when there is minimal traffic/cyclists and no HWP), this is one of the most enjoyable roads of all.
https://maps.google.com.au/maps?saddr=Pacific+Hwy%2FB83&daddr=Pacific+Hwy%2FB83+to:Pacific+Hwy%2FB83+to:Pacific+Hwy%2FB83+to:Pacific+Hwy%2FB83+to:Pacific+Hwy%2FB83+to:Pacific+Hwy%2FB83+to:Pacific+Hwy%2FB83&hl=en&sll=-33.438497,151.250968&sspn=0.02582,0.054717&geocode=FVwf__0do38CCQ%3BFahv__0d9LACCQ%3BFe9AAP4d4R4DCQ%3BFX2zAP4dXf4CCQ%3BFVRwAf4d2AIDCQ%3BFYTgAf4dkowDCQ%3BFVTjAf4dxpwDCQ%3BFcQQAv4dQngECQ&mra=ls&t=m&z=12 Galston Gorge:This has a bunch of super tight hairpins, so it's only fun when there's no one else around. It's only short, but it can be combined with an Old Pac run (see above).
https://www.google.com.au/maps/dir/-33.6801752,151.0979003/-33.6621756,151.0839983/-33.6561706,151.0677177/@-33.6640981,151.0648428,14.25z/data=!4m2!4m1!3e0 Royal National Park, AKA 'RNP' or Natio/Nasho:If you don't get stuck behind traffic then this one is really fun. There are some good/safe overtaking areas before the best part of the road anyway. This road begins as Farnell Avenue near Loftus. Most people stop at Stanwell Tops before turning back, but if you're willing to drive further south then there is a beautiful seaside drive to be enjoyed.
https://maps.google.com.au/maps?saddr=Farnell+Ave&daddr=Sir+Bertram+Stevens+Drive+to:Sir+Bertram+Stevens+Drive+to:Lady+Wakehurst+Drive&hl=en&sll=-34.208111,151.015406&sspn=0.025589,0.054717&geocode=FWBc-P0dc9AACQ%3BFS0S-P0dkfMACQ%3BFcTj9v0deoQACQ%3BFW0A9v0dfUAACQ&mra=ls&t=m&z=12 McCarrs Creek Road:Turn off Mona Vale Road and away you go. It's a great drive when there aren't too many cyclists in the way (night time can be better for this reason).
https://maps.google.com.au/maps?saddr=McCarrs+Creek+Rd&daddr=McCarrs+Creek+Rd+to:Pittwater+Rd&hl=en&ll=-33.663568,151.267061&spn=0.051506,0.109434&sll=-33.667854,151.281395&sspn=0.051504,0.109434&geocode=FUYM_v0d45kDCQ%3BFYdv_v0dSjQECQ%3BFUQp_v0d-7MECQ&mra=ls&t=m&z=14Bobbin Head Road / Ku-Ring-Gai Chase Road:Much the same story as McCarrs Creek Road (and they are also quite close to each other, if you want to do both).
https://maps.google.com.au/maps?saddr=Ku-Ring-Gai+Chase+Rd&daddr=Ku-Ring-Gai+Chase+Rd+to:Bobbin+Head+Rd&hl=en&ll=-33.670926,151.141663&spn=0.051502,0.109434&sll=-33.67664,151.134968&sspn=0.051499,0.109434&geocode=FdgZ_v0dXPABCQ%3BFThm_v0dQYACCQ%3BFYP0_f0dhmcCCQ&mra=ls&t=m&z=14 Springwood Rd / Mitchells Pass / Old Bathurst Rd:Turn onto Castlereagh Rd starting at either Richmond or Penrith, then turn onto Springwood Rd. Follow Springwood Rd uphill until you get to the Hawkesbury Lookout, where I recommend that you stop for a look at the view. Then continue until you reach the Great Western Highway, and turn left. Turn left onto Wilson Way and follow it until the roundabout where you turn left onto Mitchells Pass. Follow Mitchells Pass until the end, then turn left onto Russell St and continue along Russell St until Old Bathurst Rd, then turn left. Follow Old Bathurst Rd uphill until you reach the end (note: at the top there's a United servo with E85).
Those are the closest options in Sydney, but if you're happy to do a longer drive then continue reading...
Putty Road:It's a very long drive, but a really good one. There are multiple sections of fun and varying corners, separated by some straight stretches of road, but the best part of the drive is near the northern end just before Milbrodale, with almost 20km of non-stop awesome corners.
When you reach the northern end of Putty Rd, if you don't want to do it all over again in reverse, then you may want to turn right at Milbrodale Rd towards Broke, then turn right onto Wollombi Rd, and drive all the way to Somersby. From there, you can have even more fun by going south down Old Pac, or you can jump on the F3/M1 motorway back to Sydney.
https://maps.google.com.au/maps?saddr=Wilberforce+Rd&daddr=Putty+Rd+to:Putty+Rd&hl=en&sll=-33.114549,151.22406&sspn=1.955355,3.56781&geocode=FepX__0d2n39CA%3BFdAvBv4dek77CA%3BFQcfDf4dzwoACQ&mra=ls&t=m&z=10 Macquarie Pass:It's pretty far from Sydney, and it's only a short run, but it's so much fun you'll probably want to go up and down the mountain multiple times! You can also choose to drive down Jamberoo Mountain Rd from the top of Macquarie Pass, and/or head over to Kangaroo Valley which is also nearby.
https://maps.google.com.au/maps?saddr=Illawarra+Hwy%2FA48&daddr=Illawarra+Hwy%2FA48&hl=en&ll=-34.571249,150.654831&spn=0.050957,0.109434&sll=-34.571234,150.661195&sspn=0.050957,0.109434&geocode=FYRt8P0do0r7CA%3BFVlB8P0dtF76CA&mra=ls&t=m&z=14 Bells Line of Road:This goes out to Lithgow. It's quite a long drive, but certainly a fun one when there isn't too much traffic.
https://www.google.com.au/maps/dir/-33.572876,150.7122798/-33.541379,150.422268/-33.4728732,150.1809421/@-33.5827481,150.401997,10.5z/data=!4m2!4m1!3e0?hl=en Happy driving!