At first I wondered why you're turning at 60mph, but re-read it for the word 'curve'.
Does this happen if you take it up a bit above 60mph in other situations? 110kph (68mph) you'll start to see whole car vibration from tire balancing issues in a very prominent way. Putting more stress on one side of the car while turning could show these issues at lower speeds, which could be why you're getting that. Try [usual disclaimer] going a bit faster and see if it shows in a straight line. If it is, I'd make sure those tires are fully balanced up, not unevenly worn, and so on.
If that checks out okay, I'd guess CV joints giving a bit.
For good luck check alignment too, as it's a lot cheaper than a CV joint. Also, if you have any oversized tires (if you went wider than the stock ones which is all too common), expect and allow for slightly more vibration at higher speeds while cornering.
-M