Template not showing in my Laravel 6.6 project with Vue.js

I made the basic installation from Vue.JS in my Laravel project, this is my package.json

{
    "private": true,
    "scripts": {
        "dev": "npm run development",
        "development": "cross-env NODE_ENV=development node_modules/webpack/bin/webpack.js --progress --hide-modules --config=node_modules/laravel-mix/setup/webpack.config.js",
        "watch": "npm run development -- --watch",
        "watch-poll": "npm run watch -- --watch-poll",
        "hot": "cross-env NODE_ENV=development node_modules/webpack-dev-server/bin/webpack-dev-server.js --inline --hot --config=node_modules/laravel-mix/setup/webpack.config.js",
        "prod": "npm run production",
        "production": "cross-env NODE_ENV=production node_modules/webpack/bin/webpack.js --no-progress --hide-modules --config=node_modules/laravel-mix/setup/webpack.config.js"
    },
    "devDependencies": {
        "axios": "^0.19",
        "bootstrap": "^4.0.0",
        "cross-env": "^5.1",
        "jquery": "^3.2",
        "laravel-mix": "^5.0.1",
        "lodash": "^4.17.13",
        "popper.js": "^1.12",
        "resolve-url-loader": "^2.3.1",
        "sass": "^1.20.1",
        "sass-loader": "7.*",
        "vue": "^2.5.17",
        "vue-template-compiler": "^2.6.10"
    }
}

So I run this code that everyone does after installation npm run install && npm run dev for the mix from Laravel generate the files, every tutorial say that I must put this in the footer from my main file

<script src="{{ asset('js/app.js') }}"></script>

And always say that the file doesn't exist, it is true because when I run this command from mix above, this simple make a js and a css folder in a new public folder, this is the right approach? because I don't see any tutorial that this command makes a public folder, see the image below:

So when I explicitly put this path:

<script src="{{ asset('public/js/app.js') }}"></script>

My Vue.js is loaded and I see the default text from Vue.js in console, but when I put the template from this file TesteFood.vue

<template>
    <h1>Testando VUE</h1>
</template>

Is not showing and I make this command too npm run watch no success.