Are triglycerides soluble in polar solvents?
Polar is not a black-and-white distinction. There are weakly polar compounds (like ether), medium polar compounds (dichloromethane) and strongly polar compounds (dimethyl sulfoxide). Fats (i.e. triglycerides) can dissolve in weakly and maybe moderately polar solvents but not in strongly polar ones.

While a dipole can be induced in triglycerides to some extend, most of the long nonpolar side chain can only give weak induced dipoles. Thus, weakly polar solvents may still dissolve these by dipole–induced dipole interactions but the triglycerides are just too nonpolar for strongly polar solvents.

Water is one of the most polar solvents and additionally able to hydrogen bond. While triglycerides can accept hydrogen bonds around the glycerin moiety, they are just not polarisable enough for water to act. Hence, water prefers to form its own phase.