It makes intuitive sense that fat goes ‘right through you’ because it’s all oily and stuff, but in reality, fat takes longer to digest than carbohydrate or protein because it is not water-soluable and requires the excretion of several different compounds along with thorough mixing to be processed. Also, foods that are majority fat and protein are digested almost completely by the human body. Most of the food waste in stool is from carbohydrate (plant-based) foods which tend to be fibrous.
In the OPs case, I agree with PPs that something he is eating is irritating his colon and causing it contract, expelling whatever is in there at the moment (before the water has been removed, hence diarrhea). It’s probably a specific food or additive causing this.
Anyway, I used to have quite a few digestive problems, but now that I have figured out what foods cause me troubles and stopped eating them I have none at all. I would suggest you avoid the restaurants that give you diarrhea if the only time you have this problem is when you eat there.