That was my first experience, so I wondered why and did not know what to do, and. As the symptom was not badsevere enough to require medicine, so I did not do anything.

Just rephrasing to be more natural. I think since you have two seperate cause and effect statements, they work better as two different sentences.

I feltwas very scared to see that.

"scared" already contains the information that it's a feeling, so it's more natural to use "was" here I think.

At any rate, I thought I might have had some infection so had to take some antibiotics that I always keep in stocked up on.

After taking them for five days, the symptoms did not yet clear up, so I looked up some breast specialist clinics and I called one of them at onceimmediately.

"at once" can mean "at the same time" when combined with a quantity, so it's more natural to use "immediately" here to avoid confusion.

On the day of my appointment, a female doctor from the clinic manually examined my breasts and performinged an ultrasound for about an hour.

Since we're talking past tense, it should be "performed" not "performing"

As a result, it took about 6 hours, because there were several other patients besides me and her examinations were very thorough.

"her examination was very thorough" - just describing her examination of you
"her examinations were very thorough" - describing her examinations of you and other people.

I feltwas very scared to see that.

At any rate, I thought I might have had some infection so had to take some antibiotics that I always keep stocked up on.

After taking them for five days, the symptom did not yet clear up, so I looked up some breast specialist clinics and I called one of them at once.

On the day of my appointment, a female doctor from the clinic manually examined my breasts and performinged an ultrasound for about an hour.

She said, “iIt doesn’t appear that you have a malignant tumor in your breast and the symptoms seem to be caused by the nipple itself.”

However, an accurate diagnosis would require high-precision ultrasound and mammography, which we are unable to perform here.” She then wrote me a referral letter to the hospital where she worksed full time as a department manager.

As a result, it took about 6 hours, because there were several other patients besides me and her examination wereas very thorough.

Going Tto a Hospital

"To", being a short preposition, is generally not capitalised in a title. It's just a small issue though.

At any rate, I thought I might have had some infection, so had to takeI took some antibiotics that I always keep stocked up on.

(1) I'd personally recommend the addition of a comma here.

(2) I know that you often can omit shared words and clauses, but in this case, I think it's actually more natural-sounding to reintroduce "I" for some inexplicable reason. Curiously though, if you'd used "and", as in "I thought I might have had some infection and took some antibiotics", the "I" becomes unnecessary.

(3) "had to take some antibiotics" is fine, but it implies some necessity. Looking at the native text, the sense of necessity doesn't seem to be present, so you can also consider simply writing "took some antibiotics".

After taking them for five days, the symptom did not yet clear up, so I looked up some breast specialist clinics and I called one of them at once.

You may omit the "I" here.

On the day of my appointment, a female doctor from the clinic manually examined my breasts and performinged an ultrasound for about an hour.

She said, “iIt doesn’t appear that you have a malignant tumor in your breast and the symptoms seem to be caused by the nipple itself.

Do mind the capitalisation.

However, an accurate diagnosis would require high-precision ultrasound and mammography, which we are unable to perform here.” She then wrote me a referral letter to the hospital where she works full -time as a department manager.

"Full-time" is hyphenated and one word.

As a result, iIt took about 6 hours, because there were several other patients besides me and her examination wereas very thorough.

(1) When you write "as a result" at the start of a sentence, you're referring to the previous sentence. The implication is that the fact that you went to the hospital and was examined again by her was the reason it took 6 hours.

Additionally, "as a result" is redundant here, since "because" already establishes the causal relationship. If you wish to keep "as a result", then you must remove "because": "There were several other patients besides me and her examination was very thorough, so as a result, it took about 6 hours."

(2) "Examination" is singular, but "were" is plural. To keep the grammatical number consistent, you should write "her examination was" or "her examinations were".

There is nothing to worry about with my breast, so I will just observe for a few more months before going toseeing her in three months.

(1) Alternatively: "There is no need to worry about my breast".
(2) I'd also recommend the addition of another comma here. You use a number of conjunctions here ("so" and "before"), so breaking the sentence up with comma breaks might make it easier on the reader.
(3) "going to her" is okay, but in the context of visiting the doctor's, "seeing her" is better.