The Subaru Crosstrek gets high marks in most every review for subcompact SUVs. I'd also check out the Mazda CX3, and the Hyundai Kona. You will mostly have to look at used to stay under the $20k mark.
My daughter is talking about leasing now so that would bump up the price parameter a bit. We'll see what kind of interest rates the dealers want to offer to someone with great credit and a good job.
She may end up with a leased new car and push the buying decision back 3 years.