i have same problem with you and here is my solution:

1. You must define

   DATABASE_URL: postgresql://${DB_USERNAME}:${DB_PASSWORD}@postgres-db:5432/${DB_DATABASE}
   

   inside docker compose for backend service connect to the postgres db. here is my docker-compose file:

version: '4.0'
services:
  db:
    image: postgres
    container_name: postgres
    environment:
      POSTGRES_USER: ${DB_USERNAME}
      POSTGRES_PASSWORD: ${DB_PASSWORD}
      POSTGRES_DB: ${DB_DATABASE}
    ports:
      - "5432:5432"
    volumes:
      - db_data:/var/lib/postgresql/data
  backend:
    build: .
    container_name: backend
    ports:
      - "3000:3000"
    environment:
      DATABASE_URL: postgresql://${DB_USERNAME}:${DB_PASSWORD}@postgres-db:5432/${DB_DATABASE}
    depends_on:
      - db
    volumes:
      - .:/app
      - /app/node_modules
volumes:
  db_data:

then change the host(DB_HOST) in .env file equal to "db" (because you named postgres is "db" in docker-compose file)

PORT=3000

DB_HOST=db
DB_PORT=5432
DB_USERNAME=postgres
DB_PASSWORD=123456
DB_DATABASE=auth

the typeORM config

 TypeOrmModule.forRootAsync({
      imports: [ConfigModule],
      useFactory: (configService: ConfigService) => ({
        type: 'postgres',
        host: configService.get('DB_HOST'),
        port: +configService.get('DB_PORT'),
        username: configService.get('DB_USERNAME'),
        password: configService.get('DB_PASSWORD'),
        database: configService.get('DB_DATABASE'),
        entities: [__dirname + '/**/*.entity{.ts,.js}'],
        synchronize: true,
        logging: true
    }),
    inject: [ConfigService],
    }),