Before buying a computer, you should always make 100% sure you know what you want and
why you want it. As far as laptops vs. desktops, you should narrow down exactly why you want the laptop when desktops 100% give more bang for your buck in terms of longevity and quality.
I was in a similar position last year and decided on a gaming laptop because I wanted a computer that I could be mobile with that was also sturdier than a tablet. As I was out of school, being essentially tethered to a power cord (don't expect anything more than 4 hours on your battery life for a gaming laptop in your price range, unless you browse on lower settings) did not bother me. One plus is that newer models with more recent graphics cards seem to only fully utilize the card if the laptop is plugged in, so you can balance freedom vs power usage.
I ended up with a fairly inexpensive Asus computer
(here's last year's model, within your price range) with only one "upgrade" in installing a SSD over the fairly shitty hard drive it comes with (I highly recommend an SSD regardless of which direction you go, best computer decision I've ever made imo). I generally stay away from brand new AAA titles so I doubt it could run Witcher 3 and such well, but I play Skyrim with full mod support at max settings without issue while plugged in. Since most of my gaming time is catching up on the last several years of PC games and LoL (all usually at max settings) I am perfectly content with my laptop's graphical capabilities. If you truly want a laptop that can run brand new games at, say, settings equal to the console versions of said games, you're probably going to have to go above your maximum price.
Alternatively, you could reconsider building a desktop, if you decide it will fit your requirements. I definitely agree with Gato, if you are considering this for school you should 100% look into getting a tablet/surface/something instead, exception being if you're going in to some kind of engineering coursework that needs a real computer that's also mobile for some reason?
Last thing, I mentioned longevity in the first part, but make sure it's something you consider. If you're of high school/college age, a laptop that lasts you 4 years and no longer is not necessarily a bad thing, but if you're already working then buying something that's built to last and that you can upgrade one bit at a time can also be beneficial. Maybe today's laptops last longer, but we'll see.