You will never get the same font rendering on Linux as on Windows as Windows font rendering (ClearType) is very strange, complicated and covered by patents.
Font rendering is also kind of a subjective thing. To anyone who is used macOS, windows font rendering looks wrong as well. Apple's font rendering renders fonts much closer to how they would look printed out. Windows tries to increase readability by reducing blurriness and aligning everything perfectly with pixels, but it does this at the expense of accuracy.
Linux's font rendering tends to be a bit behind, but is likely to be more similar to macOS than to Windows rendering as time goes forward. The fonts themselves are often made available by Microsoft for using on different systems, it's just the rendering that is different.
For me, on my screens just by installing Segoe UI and tweaking the hinting / antialiasing under GNOME settings makes it really close to what Windows delivers. The default Ubuntu font, Cantarell and Sans don't seem to be very good fonts for a great rendering experience.