It’s best to use the Mac App Store and System Preferences to download OS from Apple. They chunk it into hundreds of slices, and use accelerated download sources in the CDN that can include local content caching sharing services if you set them up. This means network errors don’t ruin the entire download and speed it greatly.
Follow the guide to install older OS and it always has a link to the latest OS at the top.
Once it downloads, copy the installer from Applications folder to your external. If you don’t have enough room, copy files to the external first so you have enough space, then run the installer is my advice. If your backup is solid, just delete some apps and files until you have the 10 GB or so of space needed. Then restore them from backup once the installer is moved off the system.
You can also fetch the installer from a command line tool:
softwareupdate --fetch-full-installer