You’re not asking if normal people exist. You’re asking if happy people exist. The answer is yes, of course they do, but people aren’t static. Neither is pain and self-doubt.
Happiness (or normality or stability — whatever you want to call it) isn’t a permanent gift granted to a select and steady few. It may be found more easily for some than others, but it’s all still just a transitory phase.
Every emotional state, stable or otherwise, is impermanent. It’s all a shifting, flowing, ever-changing hot mess of pleasure and pain, neuroses and normative behavior. Happiness is fleeting, but then again, so is suffering.
Yes, there are plenty of people out there today who are emotionally stable with no underlying neuroses who aren’t secretly in a world of pain or self-doubt, but they weren’t all that way yesterday, and it won’t all be the same people tomorrow.
Your path to being among them is in recognizing that it’s not some country club that you get to join by virtue of any birthright or accomplishment. Happiness isn’t something you achieve. It’s something you discover, and it’s a discovery that can be stumbled upon again and again, depending largely on your ability to be mindful in the present moment.