Your trainer, and common wisdom, is incorrect. It is commonly thought that muscular endurance is achieved after a certain rep range. This is not true based on how substrate level respiration. Here's how it works
when you perform a repetition of an exercise, regardless of the weight, your muscles use the most available energy source, i.e. ATP. as you perform more reps, the supply of ATP runs out. this takes ~15-60 seconds, depending on the muscle type, speed of activity, and your general conditioning. But in general, any muscle that contains a high proportion of type 2 muscles will rely mostly on ATP and partial glucose metabolism.
High rep ranges (e.g. 12-20) are fine for hypertrophy, in fact, more and more research is emerging that supports high volume training for hypertrophy, as hypertrophy is a product of tension, not just a super heavy lift.
Muscular endurance is a product of training the muscles for prolonged periods of time such that they require oxygen to complete respiration. This occurs mainly in type 1 muscle, which is rich in mitochondria, where glucose is metabolized into co2 via oxygen. The complete metabolism of glucose into co2 and h2o takes a while, like a minute. So when you're running, your muscles are using ATP, but they're also using glucose, becuase it is far more plentiful and provides way more ATP and nadh for contraction. This is because running or cardio don't generate the same force during contraction.
Conversely, muscle contraction during a bench press requires the generation of force, which needs ATP. Unless you're doing a 45 seconds or more of a bench press, you're not promoting endurance. You shouldn't do more than 20 reps while lifting if bigger muscles is the goal as you need to apply a specific amount of tension.
Also, we need to define 'endurance'. I fancy myself a runner, so endurance to me is more of a cardiovascular thing. But I see what you're getting at. In that case, if you want more muscular endurance, you should be doing less pure lifting, and more conditioning and body weight exercises. You'll promote hypertrophy, but also endurance, or fatigue resistance.
Finally, here's another reason why the 15-20 reps isn't gonna prevent gains - certain muscles simply cannot handle heavy weight and low reps. Like the biceps and deltoids. These muscles are designed to move a bone or joint a certain way, and very little force is required to do that. In fact, the ideal rep range for promoting hypertrophy (and reducing injury risk) of the biceps and deltoids is 12-18.
I have a MS in nutritional immunology and am an active CPT.