Drinking coconut oil does make you poop more easily especially when your stomach is empty. An empty stomach enables coconut oil to digest faster and break down into laxative compounds that trigger bowel movement.
The more coconut oil you ingest, the stronger the urge to poop. And the faster the effect. Bear in mind that drinking too much coconut oil may lead to diarrhea.
You can mellow the urge to poop and avoid the diarrhea side effect of coconut oil by eating coconut oil with food or drinking it after food. Meaning, your stomach must have something inside to make coconut oil digest slower. And you have to have more food than coconut oil to achieve that.
Imagine you drink like two tablespoons of coconut oil but yet your stomach only has a mouthful of rice or pasta, that may not suffice to make coconut oil digest slowly.
To give you an idea based on my experience, a cup of oatmeal plus one tablespoon of coconut oil is just nice to make me poop in less than 2 hours without triggering diarrhea. Your body may respond differently, just a reminder.