It takes a lot of work to become proficient in web development. You will never reach the point where you can relax and stop learning because that point does not exist when it comes to the web. There are always new browser techs to learn, new frameworks to pick up on, new developments in organizational processes like Agile to keep track of.
My advice is to keep plugging away and learn to accept this feeling as completely normal. I love the web and I’m very good at it, but there’s no way any one person can possibly learn the whole thing, not even me.
Eventually you’ll learn enough about it to where you can develop a specialty. Specialties are niches where you can, with the blessing of employers, reduce your area of responsibility to a highly in-demand subset. Your specialty is only as valid as your ability to find jobs in that specialty.
I specialized fairly early in my career in Ruby on Rails development. A lot of devs bounce around different stacks, I stick to one. For me, trying to niche down even further would hurt my ability to get jobs, and I want to remain flexible enough to be easily employable, so I stick to calling myself a Rails specialist.
But someone who wants to keep it simpler at the cost of employability could specialize in front or back-end development. In practice most devs just let the jobs they get determine their career path.