I have noticed in my game that time really isn't that fast. I fast-forward a lot and even then it doesn't seem that fast. If I were too fast-forward a whole day in the game (24 minutes if on normal speed), It would take 15+ minutes. On The Sims 2, I could go through a whole day in like 5-7 minutes. I don't think it is only my computer because I believe others have the same problem. Is this just how the game is?

I have tried speed mods and they seem to work, but only a little. The slowness just gets on my nerves haha.

Sims 2 is also processing far fewer sims. Your lot is the only one that is active in Sims 2. In Sims 3, the whole world is active.

I suppose that means that a faster processor might speed things up, but that's probably not an option. Shutting down unnecessary processes and making sure you don't have any Sims 2 or Sims 4 packages in your game might help. But I can't really say if your speed is normal or not because I've been playing for a long time with a mod to slow things down.