Lots of good answers, many of which touch on related points, but not necessarily the ones I'd like to focus on.
We have a trainee that needs to learn a lot of tech.
How much is "a lot" here? By your own admission, it sounds like even you consider that to be a significant and non-trivial amount. Assuming you're an average person and not some gifted genius, how long did it take you to learn it all? Start with that as an anchor point.
We decided that she takes online courses with exercises independently, before touching in-house stuff.
Your approach seems very ineffective. You don't just throw everything at someone and expect them to learn it before they even know what they need to use. That's how schools often teach math and it's a methodical disaster, IMHO.
Get the trainee to work on simpler/smaller tasks, so that they learn by doing as they go. That will be productive, effective, and mutually beneficial.
Also, being a trainee implies that there should be a trainer, instructor, or mentor of some sort with the trainee's training as part of their official responsibilities. This mentor would also be accountable for the trainee's progress at least as much as the trainee him/herself.
If you're not doing this, and it sounds like you're not from your OP, then this is on you, not your trainee.
In the beginning we talked about schedules, and how long she thinks the courses should take.
You're asking the trainee to provide estimates about something they're not yet qualified to estimate. Can you really justify the reasoning and logic behind doing this to other people with a straight face?
We agreed on a reasonable schedule.
Here's something that a lot of people seem to forget: Estimates are not commitments.
Also, in what alternate universe does a, presumably experienced, logical, and rational person, think that an estimate provided by someone who is not familiar with the material will be anywhere close to reality? Again, you're asking the trainee to provide a "reasonable" (according to who and on what basis?) estimate about something that, as a complete unknown to them, they're not yet qualified to estimate.
Knowing how much there is to learn with a steep learning curve, I set aside some time in case this takes longer than thought, instead of hurrying the learning process
And the purpose of this is what? You admit that the material is both a sizeable amount and of significant difficulty; learning things like this at a satisfactory level can take months, not just "hours" or "days". It would be more effective if you look at my initial suggestion regarding a more effective learning-by-doing method. I'd avoid doing things that might unnecessarily increase the pressure on this trainee, for no good reason. Doing this will simply create more (mental) obstacles for the trainee, instead of helping.
The problem now is, that these deadlines have been moved multiple times by the trainee [...] then the same process repeats itself.
I see nothing wrong with this. It's called wishful thinking meets objective reality.
Also I learned she has registered to more online courses on her own (that is, of course, a positive thing, but doing this without discussing it first seems like an issue)
Why does it "seem like an issue"? The only concern that I think would be valid here would be if the additional material is really related to what she needs to learn or if, due to her being new, she simply thinks it's related, when it's in fact irrelevant, or at least not as relevant as other things.
Beyond this, I don't see a lot of validity to your "issue" concern.
I'm in an awkward position, not knowing if this is slacking - or if she genuinely needs more time to process what she is learning and is too scared to say she does not understand something.
I think this will depend on how much rapport and trust you've been able to earn in the meantime, what your reactions have been in your previous meetings regarding the (IMHO questionable) "schedules" that were initially set up and then updated/delayed, whether you've been expressing your frustration (either verbally or general body-language, etc.), and so on.
Try to think back and, if you have, it might be a good idea to briefly apologize, acknowledge that you should've known better when it came to the amount and difficulty of the material, and that it's ok to openly let you know if she needs something else. (Obviously, this should be done with any trainee, regardless of gender.)
I want her to learn - in a way that suits her best.
Have you asked her what, if anything, she needs from you? Have you made it clear that, as a manager, you're there to remove obstacles outside of her control from her path rather than trying to spend your time micromanaging people?
Anyway this looks bad for me, not being able to manage this trainee and be any kind of an authority (letting a trainee dictate schedules)
What looks bad on you is that you should've known better regarding the amount and difficulty of the material. I once had a friend co-worker who severely underestimated the complexity of a task, and gave it out to an intern. Not surprisingly, the intern failed to accomplish his project. It lead to an overall negative and frustrating experience for everyone involved, including myself who as (unofficially) acting as the mentor to the intern's mentor.
I see lots of similarities here and I'm saying these things because it seems like you, as the more experienced person, need a reality check. You clearly seem to be holding yourself accountable, but you should do that not because it makes you "look bad", but because it makes you an ineffective manager and leader (there's a difference).
Also, if you really "want her to learn - in a way that suits her best", then you should provide a reasonable amount of trust and let the trainee openly communicate what their actual needs are. Even if this person needs something (e.g. more time to understand, etc), previous interactions (or a lack of them) might be part of the situation here.
How do I regain control of the situation?
You cannot "control the situation" while at the same time claiming that you "want her to learn - in a way that suits her best". You're trying to eat your cake and also have it; can't have it both ways.
I also don't know what "controlling the situation" would even mean, as you can't make her, or anyone else, learn/understand lots of complicated things at a faster rate. But perhaps some of the points above (e.g. learning by doing, smaller/simpler tasks, having an actual mentor, etc) might be of help to both of you in the process.