The Bad News
High frequency, high-load ab and oblique training may widen your waist. No big deal if you're after the strongman look, not so great if you're a competitive bodybuilder or figure/bikini competitor... or just want to look like one.
Consider the case of 47-year old competitor Alice Galea:
Five to six days per week she would do weighted cable crunches, weighted leg raises, decline weighted sit-ups, oblique crunches, and plenty of side planks with hip pikes and plank variations. Her waist grew from 24 to 28 inches! She gained 4 inches around her waist from the high frequency, high-load abdominal/oblique work. She temporarily stopped all ab and oblique training to fix the problem.
The Good News
Several studies show that squats and deadlifts don't activate the rectus abdominis, internal obliques, and external obliques to high degrees, and those are the muscles that give you a thick waist. Most popular ab and core stability exercises far out-activate the abdominal wall muscles when compared to squats and deadlifts, which is perfectly logical from a biomechanical perspective.
Those who want a narrow waist can enjoy their squats and deadlifts without fear of developing a "blocky" midsection, something many bodybuilders once worried about. Those who want a narrow waist would be far better off doing squats and deadlifts than many of the traditional "waist-slimming" movements like weighted cable crunches, decline weighted sit-ups, weighted oblique crunches, weighted leg raises, planks, etc.
So What Should You Do?
Want abs that aren't overly developed to the point of making your waist thick? Focus on diet and gaining overall strength with various lower and upper body movements. Your abs will show and won't be hidden under a layer of fat. Nor will they be excessively developed.
Sure, do a few sets for abs occasionally without fearing overdevelopment, but don't focus on progressive overload (always trying to go heavier and heavier) and don't go to utter failure day after day.