My character needs to go see a psychiatrist, and as I am someone who is not that knowledgeable about the way psychiatrists talk to their patients, or manipulate them to get answers, or advice them, I don't believe I can write a good conversation between the two of them or at least a realistic one, but I also believe that it is a necessary scene for the plot. How can I peacefully skip this scene without making the story imbalanced and without making my readers feel that I deliberately skipped it because of my somehow poor writing skills?
If a scene is necessary (which is the word you used), then no, you can't skip it.
Or at a minimum, if you did skip it, you would need to summarize for the reader what happened in that scene (which is basically still not skipping it). And to summarize it you need to know what actually happens in that scene.
So either way, you need to do the research necessary to figure out how that scene plays out.
Bottom line: if a scene is integral to your plot, you need to know what happens in it. Otherwise how would you know what the ripple effects from that scene should be?
Don't worry about getting it perfectly right the first time; do a little research, try it out, then give it to beta readers to get feedback, and if it needs work just iterate on it. You'll get to something realistic eventually. You got this.
If there is a problem in the story, the problem is earlier in the story.
In your case, you are asking if not writing the psychiatrist scene will make it look like you avoided writing the psychiatrist scene. It might look that way if the story promises (earlier) that the psychiatrist scene is an important plot point.
So, give the reader something to scratch that itch, before the meeting with the psychiatrist. (Alternatively, signal that the meeting is not important, earlier.)
Obviously we don't know the details of your story, but imagine that the hero of the story needs to see the psychiatrist because she thinks this woman is having an affair with her husband, and she wants to confront her and ask her a few questions. In that case, the reader very much wants to be in the room for the conversation. So the author, if they are unwilling to provide that scene, will need to satisfy the promised reveal in some other way. Example: The hero of the story might imagine all the ways the interaction could go, for example. Since our hero is not a psychiatrist, their imagination will not have psychiatric expertise. I could imagine this approach working, although it feels tortured. Then skip the scene with something like: "After the session, (I wanted to kill him)/(I was so relieved)."
Alternatively, imagine the hero is being forced to see a psychiatrist, and is terrified of what he will be asked. In that case, again, the reader expects something--but you could write the anticipation of how bad it might be, prior to the meeting. Maybe the hero starts to imagine chains, and a dungeon, or something hyperbolic like this, in the doctor's office. Perhaps the hero resolves--again, beforehand--to not say a single damned word. Have him build it up in his mind, make it as extravagant as you like, to satisfy the reader, and then jump (skip) to: "It turned out, the shrink just wanted my dad's insurance information."
If the meeting is signaled (earlier) as important to the plot, which I think we're all assuming here, then you need to satisfy your reader. That's the basic principle to follow, and if you sense that something isn't working, look earlier in your story to play with how you had set it all up to reach that point. The more you write, the more you will see ways to change storylines. There's too little information in your question to give a more solid answer than that.
TL;DR: Yet another example of how "Write what you know" is the worst advice you can give a new writer.
To quote a beloved character from the 90s-2000s era cartoon, "YOU MUST DO REEEESERCH!"
That is the best advice to skipping a critical scene where your character is talking to a psychiatrist because you have no clue how psychiatrists talk to people is to... talk to a psychiatrist about how they do their job. Present the scenario to them. Hell, ask the shrink if you can role play out the scene... you be your character and interact with the psychiatrist... and then write that scene in your book.
You'd be surprised what accesses you can get if you tell people "it's for a book!" after you ask them your questions. Do not let a lack of experience stop you from writing important scenes.