Yes Minecraft itself is split up in a server and client aspect, even in singleplayer. This has nothing to do with the modpack.
It is not this separation that is making you lag. You have something that is taking up taking up so much processing time that the server aspect cannot process all the things in the world that it needs to in the time allotted. Number one cause of this is usually mobs AI. Look for lots of mobs concentrated in one spot(animal farms, villagers breeding out of control etc.). Another cause is bad and overuse of machines/magic/automation. Everything that does something needs to be processed and handled. If you set up a gazillion units that does something at ridiculous speeds, then that is going to significantly add to the processing time needed per tick. Once it takes longer to process a tick than a tick lasts: Lag.