I don't know why people keep recommending Macs for games. They are overpriced and not suited for the task. Here is a high performance laptop within your budget and will handle most games including Skyrim and Fallout 4.
It is an Acer Nitro 7. High performance in a thinish and smallish package.
Windows 10 Home
Intel® Core™ i7-9750H processor Hexa-core 2.60 GHz
NVIDIA® GeForce® GTX 1650 with 4 GB dedicated memory
15.6" Full HD (1920 x 1080) 16:9 IPS
16 GB, DDR4 SDRAM
512 GB SSD
I have multiple Acer's including two Predators, a V-Nitro and several Aspires. So far out of the 23 Acers, one had a mainboard failure last year but it was a machine with a Core 2 Duo CPU so it served it's life. I had and unfortunately still have Lenovo's but will never buy their products again or until they prove they aren't a disappointment. Had two Yoga Y7070 laptops, both died in 5 months, Have 3 E73 towers, one had a mainboard replacement, another a PSU replacement and the last has been reinstalled 4 times to fix random glitches. Also have a couple G series and B series laptops all with a problem or another. Got 5 Dell's too. All but one run quite well. The one that doesn't has an issue with the display constantly turning off with no solution. Been doing this for over a year. I also have a few custom built towers and had some HP's. The HP's I didn't have good experiences with and the towers range from Celeron's to i9's.
You may wonder why I have so many machines. I own a business that requires all of this.