Sounds to me that you might just need to relax and not try so hard. Accept her as she is, nervous or not. Being nervous is a part of what introverted people are. When she's nervous, allow her to be nervous while you tell her that you know she can do this. Then allow her to do it and perhaps fail. Stop trying to make the world perfect for her.
As to strengthening your own sense of self, be positive when you talk to yourself. Remind yourself that you don't have to be a perfect parent just because you're a therapist. I wonder if she's perfectionisitic in part, at least, because you are perfectionistic. I suggest, as you suggested, that you're too close to your daughter to be realistic in assessing the way that you parent. I would guess that you are on top of things with her but you lack confidence. That is just what comes thru for me from reading your post.
You know how to strengthen her sense of self. There are no tricks for doing it. Acceptance as she is, praise, and all the other things you wrote about doing. It's OK to be nervous. As she matures she'll gain more confidence from having the successes you're setting up for her and which happen in her life.
You calm fears just by being there. Hold her hand. Tell her you know she's nervous and you know she'll be just fine. Focus more on the expected success than the fear.
If staying in the room without you staying is a concern I suggest the first step is for you to know that she can do it and that you don't need to protect her by staying. She will overcome the fear much quicker if you aren't giving her the message that you're worried about how she feels.
Be matter of fact. One way to teach this sort of independence is to talk with her about how you're going to do a new bedtime routine. Do the "bath, story, hugs" ritual. After you tuck her in you're going to leave because you know that she is alright. Tell her you will come in and check on her in 5 minutes. Go back out. Then wait 10 minutes and go back in and check on her. Increase the length of time in between checks. Just give her a pat or in some minor way acknowledge your presence but don't get sucked into tears and stay. Leave, say you'll be back and then go back. It may take several nights of doing this. Eventually she'll realize that you are still there and will take care of her.
I suggest you have to show her by leaving and staying gone for a length of time that you are confident that she can do this. When you stay with her you are telling her she's right when she thinks she can't be alone.
I suggest that when you have confidence in yourself and in her that she will be less fearful. One of my well used phrases for myself is to fake it until I make it. That is probably what will help you in regards to leaving her in her room. You can do it!
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