It should, but it doesn't. I get 20-30 fps in WoW. I can actually play it in 4k with a 1060 gtx and still not push the card to it's fullest. And yet, whether the resolution is 1080p or 4k, the framerate is the same.
If you have the 6GB version of GTX 1060, then you should be able to run it well, even at 4K. The 3GB version will probably spill into main RAM at 4K, especially if you go bananas with the other graphics settings.
If you have 16GB of RAM, I wouldn't upgrade. You'll need CPU+Mobo+RAM+Cooler, and the performance of current platforms does not justify the price in my opinion, unless you're doing something very well threaded (rendering, compiling, transcoding, etc)
What I would do, is buy an i7 3770. Your mobo will support it out of the box, but I would apply the latest pre-2018 Bios update anyway. With ~$100-$150 you'll be done.
I'm playing WoW with i7 3770 and an RX470 at 1080p but rendered at 4K and down-scaled (that in-game setting for 200% resolution) and am pretty happy with the performance and everything looks gorgeous. Just to give you an idea, in Boralus right outside the inn on the evening after the weekly reset I'm getting 35-40 fps, but I only noticed this when I ran Fraps because it's smooth (probably Freesync helps). In the wild I'm getting 50-60 fps. I haven't run Fraps during raid or dungeons, but I only noticed a little stuttering very occasionally in Uldir 25-man, and this is with many addons alerting me. I've noticed slowdowns when I back into a wall and the third person view practically becomes first person and there are many spell effects, but I quickly reorient the camera angle.
If you're interested, I can post here my WoW graphics settings. I found the main CPU resource waste is the view distance and environment detail. Mine defaulted to 7-7 but didn't like the responsiveness and lowered them to 6-6 and it's much better. They only influence the rendering of things very far, like far away mountains won't have features, just a shape.