Prepare blue water in beaker.

Using a funnel, fill the U-tube with (colored) water up to about one-third onto the lined area.

Add oil (using the syringe) on one side. One needs a good amount of oil -- one or on-and-one-half syringes full (maybe up to a height of 5 or 6 cm).

What happens: The pressures at the interface between liquid and air is for both sides 1 atm, and the pressure at the interface between oil and water is the same as the pressure in the water at the same height of the water-only column. The increase of pressure between the air and the previously-described point in the liquid is density of the liquid (either oil or water) times g times height of liquid column. Since the density of oil is smaller than water, the height of the column of oil will be larger. That means the side with the oil will have an overall larger column than the side with the water. One only sees the effect if there is enough oil on top of the water column.