My problem is that I struggle with finding/posing questions that are worth researching.
I think the reason is my lack of experience, but I am eager to learn and don't want to only rely on my supervisor.
Hmmm. Muddle. Try. Fail. Muddle. Repeat. Repeat. Try. Succeed (partially). Muddle. Refine. Muddle. Succeed (maybe).
To paraphrase a bit of wisdom from another field: Posing good questions comes from Experience. Experience comes from posing bad questions.
There isn't a plan in mathematical research (or in most other fields) that will guarantee success. You are moving into the unknown, which is, well, unknown.
But there are a few hints that work (sometimes...partially). First, you need to deeply understand the essence of what you are about. This requires insight and insight isn't always gained even by earning a doctorate. Insight is what knocks you off the donkey when you aren't expecting it. Insight is understanding something one moment that eluded you the moment before. Insight sometimes comes from hard work, but it also sometimes comes from just riding on the donkey to get somewhere.
Read a lot of papers.
Think about possible extensions of those papers.
Think about possible variations of those papers.
Think about applying the method of proof of those papers to some other, possibly unrelated sub-field.
Take a lot of notes and update them when you have any small insights. I find it useful to carry some of those notes around on index cards, just to refresh my memory (on the donkey).
Think hard about the essence. What is the derivative? What is the essence of the derivative? Why is the definition like it is? Are all of those parts essential? Can they be changed? What would I learn if I change something there? Derivatives work nicely on nice functions. What if the functions aren't nice.
Talk to a lot of people. Share ideas, don't hoard them. Work together. Ask "what if".
Take risks in your thinking. New sub-fields of math have often arisen from "risky thinking". Algebraic Geometry? What a wild idea.
Try again. Make notes.