With the disclaimer that I don't get paid for my Typescript work (I use it for some personal projects, as it saves me a bit of time compared to C++), I have this as the compile scripts entry in my package.json:
"compile": "tsc -p ./tsconfig.json",
I also have a start script, for running it from the command line: "start": "npm run compile && node ./.build/main.js"
tsc has some kind of file-watching mode, that I assume compiles when you save stuff, but I don't use that, because I'm not a monster.