You can turn off certain effects; specifically, those generated by players. Not those generated by NPCs, enemies or the environment.

In /characterconfig under Character you can find these settings. Alternatively, you can use the / commands.

/bfx self all
/bfx party simple
/bfx other off
/bfx enemypc off

This will allow you to see your own effects in full, your party's effects only display if they are important (healer bubbles or damage areas, which you are meant to stand in), and no effects from anyone else. The last one disables effects from others in PvP.