I know core has already got custom fees, but also I think there should be a thing where it fetches some info from a website and then it changes its recommended fees to suit the current market...
Why should core check a third party website for the fees? It already has its fee estimating, and if you actually have it run as it is supposed to, I think it is fairly accurate. In order for it to be accurate, you need to let Bitcoin core run for a while so that it receives enough blocks to do an accurate estimate.
And also - it should show you the calculated fee BEFORE you click SEND.
It does show you the recommended fee/kB. If you use the coin control feature, then I think it will also tell you how much you should expect to pay in fees.