My English teacher had advised me to find a website where my texts could be reviewed by native speakers, which would help me to improve my writing.

recommended instead of advised would be more natural here I think

After a few hours using it, the website completely reversed this initial feeling.

After a few hours using it, my initial opinions changed

Focused on the goal to better achieve it!

It allows you to focus on your goals and acheive them

Last week, I discovered the website LangCorrect, and, honestly, the resultwebsite exceeds my expectations!

It isn't incorrect to refer to your discovery as a result (of your search), but it might be a little bit of a mismatch if it is the website that exceeds expectations. Let me see if I can think of an analogy: "I discovered a new French patisserie, and the result exceeds my expectations." I think it works better to say the pastry or the site exceeds your expectations. What do you think?

My English teacher had advised me to find a website where my texts could be reviewed by native speakers, which would help me to improve my writing.

"to help me improve my writing" is an alternative wording to "which would help me to improve my writing."

At first sight, I was slightly disappointed, as the website didn’t look quite modern [or even serious enough].

I'm unsure what you mean by "serious enough." Perhaps it didn't seem "rigorous" or "academic"?

After I used it for a few hours using it, the website completely reversed thismy initial feeling.

This is an advanced grammar topic. I'm not sure I can explain it. The first part of the sentence "after a few hours of using it" actually refers to you, the user, so there are two different subjects. I like balancing the active voice "I used it" with "the website reversed."

Oh, a funny point that I didn’t expect neither, the counterpart — reviewing others’ texts — is really interesting.

"didn't expect" paired with "neither" is a double negative which is a no no in English. You could replace "neither with "either," but that word is not necessary, and there isn't a good reason to have it there.

Even though I often struggle a lot to explain why something is unnatural in my native language!

Grammatically speaking, this is an incomplete sentence. It's a dependent clause masquerading as a sentence. The way you can tell is to remove "Even though." Now you have a complete sentence: "I often struggle a lot to explain why something is unnatural in my native language!"

So, grammatically, you should put a comma after "language" and add a main clause. You could fix the situation by adding this sentence to the one before:

"Oh, a funny point that I didn’t expect, the counterpart — reviewing others’ texts — is really interesting, even though I often struggle a lot to explain why something is unnatural in my native language!"

This makes a rather complex sentence, but I think it works grammatically. Or, you could see if your not correct sentence escapes detection by the grammar police.

Last week, I discovered the website LangCorrect, and honestly the results have exceedsed my expectations!

My English teacher had advised me to find a website where my texts could be reviewed by native speakers, which would help me to to help improve my writing.

When I finally decided to follow her advice, I looked for ideas on Reddit, and. I instantly stumbled upon highlythe positive recommendations for LangCorrect.

At first sight, I was slightly disappointed, as the website didn’t look quite modern or even serious enough.

After a few hours using it, the website completely reversed this initial feeling, I felt better.