React app not loading with docker-compose

The Dockerfile for my React client:

FROM node:10

WORKDIR /app/client

COPY ["package.json", "package-lock.json", "./"]

RUN npm install --production

COPY . .

EXPOSE 3000

CMD ["npm", "start"]

The Dockerfile for my Express backend:

FROM node:10

WORKDIR /app/server

COPY ["package.json", "package-lock.json", "./"]
RUN ls
RUN npm install --production

COPY . .

EXPOSE 5000

CMD ["node", "server.js"]

My docker-compose.yml file in my project's root:

version: '3'

services:
  backend:
    build:
      context: ./backend
      dockerfile: ./Dockerfile
    image: "isaacasante/mcm-backend"
    ports:
      - "5000:5000"
  frontend:
    build:
      context: ./client
      dockerfile: ./Dockerfile
    image: "isaacasante/mcm-client"
    ports:
      - "3000:3000"
    links:
      - "backend"

My server.js file under my backend folder:

var express = require("express");
var cors = require("cors");
var app = express();
var path = require("path");

// Enable CORS and handle JSON requests
app.use(cors());
app.use(express.json());

app.post("/", function (req, res, next) {
  // console.log(req.body);
  res.json({ msg: "This is CORS-enabled for all origins!" });
});

// Set router for email notifications
const mailRouter = require("./routers/mail");
const readerRouter = require("./routers/reader");
const notificationsRouter = require("./routers/booking-notifications");
app.use("/email", mailRouter);
app.use("/reader", readerRouter);
app.use("/notifications", notificationsRouter);

if (process.env.NODE_ENV === "production") {
  app.use(express.static("mcm-app/build"));
  app.get("*", (req, res) => {
    res.sendFile(path.join(__dirname, "mcm-app", "build", "index.html"));
  });
}

app.listen(5000, function () {
  console.log("server starting...");
});

When I run:

docker-compose up

I get the following output in my terminal:

$ docker-compose up
Starting mcm_fyp_backend_1 ... done
Starting mcm_fyp_frontend_1 ... done
Attaching to mcm_fyp_backend_1, mcm_fyp_frontend_1
backend_1   | server starting...
frontend_1  |
frontend_1  | > [email protected] start /app/client
frontend_1  | > react-scripts start
frontend_1  |
frontend_1  | ? ?wds?: Project is running at http://172.18.0.3/
frontend_1  | ? ?wds?: webpack output is served from
frontend_1  | ? ?wds?: Content not from webpack is served from /app/client/public
frontend_1  | ? ?wds?: 404s will fallback to /
frontend_1  | Starting the development server...
frontend_1  |
mcm_fyp_frontend_1 exited with code 0

My backend exits with code 0, and I can't load my app. My backend is running though.

What am I doing wrong, and how can I get my React-Express-Node app running with Docker Compose?