Does world size affect multiplayer lag?

I am trying to play multiplayer with the steam integration for multiplayer, but we have a small amount of server lag on our current large world.

Would restarting on a small/medium world help the server lag?

No, but Yes and No.

What is commonly referred to as 'lag' is the latency between client and server, or in other words, the correlation of the time it takes for a package (data) to be traded from the server to the client. This does not change when you have a bigger package (In this case: World Size)

But Yes, while lag doesn't increase, bigger packages will take longer (there's more data to transfer, albeit the speed doesn't change. Same speed, more data = More Time).

And no, the time isn't really going to increase. In theory the time does increase, however in practice, unless it's a ridiculously huge package, you won't even notice the transfer taking a milisecond longer

In other words, having a smaller server will cause less time to be used to load things, but it's so small that it will barely be noticeable

From what I have played on multiplayer Steam Terraria, the world size doesn't seem to have any effect on lag... it may just be the distance between you and your friend or a glitch. I've had a glitch much like a lag thing were when I joined my friend's world: I would get kicked out about 20 seconds into the world, and I lagged a lot to, but when he joined mine it was perfectly fine... and another one of his friends complained about this to him, so maybe just try to join the opposite player's world. Oh, and it may also depend on how old your computer / Terraria version is.