React Router not loading page

I'm following an online React tutorial. In the tutorial React Router 3 used while I got React Router 4 when I downloaded React Router (and react-router-dom). The code in the tutorial looks like this.

import React from "react";
import {render} from "react-dom";
import {Router, Route, browserHistory} from 'react-router'

import {Root} from "./components/Root";
import {Home} from "./components/Home";
import {User} from "./components/User";

class App extends React.Component {
    render() {
        return (
            <Router>
                <Route path={"user"} component={User}/>
                <Route path={"home"} component={Home}/>
            </Router>
        );
    }
}

render(<App />, window.document.getElementById('app'));  

I'm trying to rewrite the code to work with React Router 4 like this:

import React from "react";
import {render} from "react-dom";
import {Switch, BrowserRouter, Route} from 'react-router-dom'

import {Root} from "./components/Root";
import {Home} from "./components/Home";
import {User} from "./components/User";

class App extends React.Component {
    render() {
        return (
            <BrowserRouter>
                <Switch>
                    <Route path={"user"} component={User}/>
                    <Route path={"home"} component={Home}/>
                </Switch>
            </BrowserRouter>
        );
    }
}

render(<App />, window.document.getElementById('app'));

The code for User.js looks like this:

import React from "react";

export class User extends React.Component {
    render() {
        return (
            <div>
                <h3>The User Page</h3>
                <p>User ID: </p>
            </div>
        );
    }
}

When I try localhost:8080/user it returns nothing.

I only get

<div id="app">
    <!-- react-empty: 1 -->
</div>

What is it I need to change in my code?