Basically, they don't get on well together, for various reasons. I don't really understand it, but then again, I am looking in from the outside (so to speak).
There's part of the problem. Not that I am insinuating that you are the problem, au contraire, I imply that you have found a facet of the issue at hand. As dungeon/game masters, we're more than rulebooks - we are the arbiters, mediators and overall facilitators. Granted, this is a subjective statement mind you, but as far as I and several other GMs see it, it applies. If you can't/won't get to the bottom of the issue through compromise or collaboration, since avoiding isn't working you need to take an authoritative stance.
There's a piece of the question lacking, IMO, and it's what this conflict between them stems to. Since this isn't a problem solving site, I'll address that as follows - if they aren't willing to keep their fight out of your/their/the rest of the group's fun, then it's purely-and-simply spoiling fun. Don't bother with a campaign that's just being treated this way, just like you wouldn't bother with baking a cake for people you know will end up playing target practice with instead of eating. It's disrespectful to you, to each other and to those who aren't involved in the conflict. The point here is to have fun. Since they (or at least one, from what I understand of the question) are there to harass the other player and not play the game, boot them if they don't start being respectful.
The problem crossed over to my domain when one player started issuing in-character death threats. He tells me he was only joking, but he continues to make them regularly although I've asked him to stop.
Further evidence that they're disruptive to the game, and now not just to other players but to you.
I understand that they don't get along, but how can I stop them from arguing during the sessions I am running?
DM mandate is one option, discussed above. Seems it's the option you're being forced into through the problem player(s) actions. Will suck to do, but think of how much worse it would be if you lost all four players instead of just the one or two. I've abandoned games where this was happening and the DM avoided it.
Furthermore, what has your experience been in similar situations? I am looking for experience not opinions.
Run across this a couple of times as a DM and handled it rather bossily. On a do-over, I would have run a few less confrontational methods. Currently, two of my regular players who get into fights are playing together (exclusively) in an evil campaign so that they can openly quarrel in character and blow steam that way. So far it's working, but they don't sound as explosive as what you are describing.
Ultimately, it boils down to how much you're willing to bend, how much you and the other players are willing to tolerate, how well you know those involved in the conflict and how much you want to game with these people, specifically.