Are you familiar with defining boundaries? A part of defining boundaries is to decide what you will and will not accept. You are the mother and have the final say in what your kids will be allowed to do and what the consequence will be if they are not following your rules.
In this case, it's quite fair to seperate the kids. Separation is a natural consequence of not getting along. They can come back out when they decide to get along. You'll need to tell them specifically what you expect from them and give them options. For example: share toys, play with different toys, play apart. It will help if you can give each child something different to do once they come out of their rooms, corners, where ever you've put them so that they cannot see each other.
Or you can give them a specific amount of time to be seperated. 5 minutes sounds like a good amount of time for this age.
They will test you to see if you really mean what you say which means that at first you will have to pay close attention to what is going on.
Some times you'll want to send just one child to his/her room. For example if the 4 yo insists on helping the 2 yo who does't want to be helped you could send just the 4 yo to her room or even to another part of the house to do something else. Being separated does not mean that they can't do something else. The goal is to stop the bickering.
If you're like me you may be spending too much time trying to teach them how to get along. At this age, short simple requests work best. For example: brother is playing with that toy now. You find a different toy. Or vice versa.
When both are fighting over the same toy you can also take the toy away and put it up. Then both will find different toys. If they fight over the next toy, put that one up.
Because he 4 yo understands more than the 2 yo and probably has learned some social skills at school you can talk differently with each one about your expectations (rules) and the consequences that you will give them each time. When both are bickering be sure to give them both the same consequence. If you don't they'll have another cause for bickering.
Whatever you do, do not become a part of the bickering by yelling at them. Separate them before you get emotionally involved.
You could also have a boundary rule that requires the 4 yo to stay away from the 2 yo when she first gets home. Since you've been with the 2 yo already, I suggest you spend some time with just the 4 yo after school.
It is definately fair to separate them. It is fair to require that they play in different rooms or areas of the house. Once you are able to remain calm the situation is very matter of fact. "We do not fight over toys. In this house we share. If you can't share then I'm putting the toy up." another example; "If you can't play nicely with each other then you'll need to play separately." Then comes the tricky part. Follow up. Be sure to take the toy or to separate them immediately when they continue to fight.
I'm assuming that you've already tried teaching them how to share. I've found that frequently I can suggest a different toy or a different way of playing that will help them get past the "I want it! You can't have it stage."
This is not easy to do. First you have to be confident that you are in charge and that you have the right to have peace. They also have the right to have peace.
Kids bicker. You can't prevent it but you can reduce it by having definite boundaries (rules) that you consistently enforce.
They will also behave better if you're able to spend time with them showing them that each of them is important to you. Modeling for them how to treat another person with respect. You could role play sharing at a time that they aren't bickering.
You have my sympathy. My grandchildren frequently get into it with each other. The oldest does want to "help" the youngest. Actually I'd call it boss the youngest. There are built in dynamics related to birth order.
I dread driving with both of them in the same car. Circumstances gave me help. I had to buy a new car and chose a SUV for a variety of reasons. One of them was that they could sit in two different rows. That has helped.
I have had times when I've been at their house "baby sitting" when I've caught myself saying over and over in an increasingly more frustrated tone to "stop it." That just doesn't work. I have to follow up my demand with a consequence when they don't stop. Separating them seems to me to be a logical way to handle it.
Once you've been able to get the bickering under some semblence of control you can then work on any issues that cause the bickering such as it's usual for the older to boss the younger one around. With understanding and practice you may be able to reduce some of that bossiness.
With my grandchildren, it's obvious that they are jealous of each other. I have to be careful to give both of them attention while I'm with them. And to listen to each one's complaints about the other. I just listen. I don't try to change their minds. I feel successful in my efforts perhaps 60-70% of the time.
Both my daugher and I rely on separating them before we get so frustrated that we become part of the bickering. When I'm busy with something else or feeling overwhelmed I don't try to do anything but to keep them separated.