Indiana Jones travels to Istanbul in the next part of the adventure. You'll find the Royal Shield artifact as you walk down the hallway in the exhibit area, break the armor statue with your whip or by pushing an enemy into it. Take the armor pieces and beat two henchmen down to gain the "Vandal" Glory Move. This area is hard, so keep using the broken armor to take out the rest of the bad guys. Once the fight is over, pick up the armor helmet off the ground and place it on the middle bust figure. Right after you cross the bridge, turn left and you'll see the Royal Gauntlets Artifact sitting in plain sight. You'll have to shoot your way through this next part and then you'll find the Gold Belt Artifact on the balcony. Push the bust statue in the courtyard to enter the next area.
Once you break through the wall into the waterfall area, you'll see the shining hat in the distance but you won't be able to retrieve it just yet. Climb up to the ledge to the left and push down the statue to smash it. Push the resulting block to the other side and use it to hoist yourself up to the ledge you can shimmy on. Grab a skull from the pile of skulls and jam it into the gears, using an attack swing. Jam the next set of gears to quickly grab the Gold Relic artifact sitting in plain sight. Drop down and shimmy over to the right, push the standing coffin down to enter a secret chamber with the Holy Book Receptacle artifact. Once this is done swing across to the other side using your whip.
After the next set of gears (before you swing across to the other side using your whip), jump down to the opening on the right. Hop up to the spot where you can shimmy across to grab the Crystal Bottle artifact. Right after you escape the rotating blades, you'll come to the third mosaic. Look in the burial chamber to the right and you'll find the Royal Crown artifact. Now, use your whip to climb up the wall to the left of the mosaic.
At this point in Indiana Jones and the Staff of Kings, you will reach the spa room. Pick up and carry the three busts and place them on the appropriate pedestrals. This is with the male in between the two women. Head down the crumbling ledges and just keep going down until you reach the ground.
After the first timed jump across the spinning gears, turn right and hop the small wall. After the next timed jump, you'll find the Dagger of Topkapi artifact sitting in plain view. There is one more timed jump. Head up and push the statue down into the rotating gears below. Do a couple of whip swings and you'll see a shining hat in the distance. You will have to settle for the view for now. Turn right and head up the stairs and you'll find the Gold Flask artifact.