As a rule of thumb the concrete, low-level help from supervisor to student is inversely proportional to the seniority of supervisor and student. A bachelor student will need more help while a PhD student will be expected to work largely independently, for example.
Crucially, there's a point at which "you're the expert now" and there simply is no-one else to ask. Much of university/higher education aims at getting you there.
So it is usually not really about how long it takes an expert versus you to get the job done. Most of the time, it's about how long it takes you to learn the skill to get the job done; getting the job done is a nice side-effect and learning opportunity but not necessarily the goal itself.
In this regards, try and see both experts and supervisor as tools – just like a book on the topic, or Google, or forums, or peer groups, or courses, or similar. Notably, human experts and supervisors are a much more limited resource than, say, an instant search engine 24/7 available. As a rule of thumb, your first instinct should always be to start at the more available resources and move to more expert resources as a) your grasp of the topic increases to narrow the issue down and b) the necessity for expert advice to solve the issue becomes clear.
So yes: You should try and do it on your own. Only if that fails, then go ask an expert.
That said, your supervisor should not just be an elusive mastermind that you do not dare talking to. Learning to work independently includes learning when you are at your actual limits. Your supervisor might want you to push your limits to improve them but the point is not to ignore those limits. So:
This attitude of his makes me anxious and self-doubting even more. Now I hesitate every time I want to ask him something.
No matter where you are in your studies, this is something to talk about with your supervisor. That is not about putting blame on them or you, but about making you learn how to deal with such a situation yourself. Be honest, be upfront, ask for their advice and make them aware there is a problem that keeps you from working independently.
Ultimately, your supervisor is not there to give you a fish. They are not there to teach you how to fish. They are there to teach you how to teach yourself how to fish.