I know you said you don't want to use setInterval, but it's needed here if you want to do this without libraries. Fortunately, I have a feeling you meant setInterval with 14 days as the interval, and that's not necessary. Here's what I would do (sorry I can't really test it before answering because that would require about a month):

const startDate = new Date(x); //x is just whenever you want to start this: if you
//don't want to use a backend with a database, you'll need to hardcode this in the javascript

window.setInterval(()=>{
  let date = new Date();
  let twoWeeks = 1.2096e+9 //2 weeks in milliseconds
  switch (Math.floor(((date - startDate)/twoWeeks)%4)) {
//difference between the dates, divide it by the twoWeeks number to get how many of
//that interval have passed, get the remainder from dividing by 4 in case more than
//8 weeks have passed, then floor it to get an integer
    case 1:
      function1();
      break;
    case 2:
      function2();
      break;
    case 3:
      function3();
      break;
    case 0: //will happen if it's happened 4 times or 0
      function4();
      break;
  }
}, y); //y is any interval in milliseconds, shorter will be more precise but will use more memory

if you want to prevent it from doing function4() before the first 2 week interval passes, you could update the function like so:

window.setInterval(()=>{
  let date = new Date();
  let twoWeeks = 1.2096e+9;
  if ((date - startDate)/twoWeeks < 4) return;
  switch (Math.floor(((date - startDate)/twoWeeks)%4)) {
    case 1:
      function1();
      break;
    case 2:
      function2();
      break;
    case 3:
      function3();
      break;
    case 0:
      function4();
      break;
  }
}, y);

Note, this will have the functions be called more than once within that interval, but considering it's just setting which div will be visible, that won't be a problem.

One way of solving this is by using cron.

The docs in that link explain it well but essentially you can set a date and time and pass in the function you'd want to call based on the given date/time params

var CronJob = require('cron').CronJob;
var job = new CronJob('* * * * * *', function() {
  console.log('You will see this message every second');
}, null, true, 'America/Los_Angeles');
job.start();