I really want to improve my writing in English

I have been paying attention to improve improving my listening and speaking but recently I really want to improve my writing. However, I have no idea about how to achieve it at present. I talked about it with my friend today and he told me that grammar would be more important in writing compared with speaking.

At the very start, I was told not to pay too much attention on to [1] grammar in order to encourage me to open my mouth and to eliminate my hesitation. But now I have a very big grammar problem. I would always feel confused by some expressions. I am so afraid of making mistakes when I need to write something. My friend told me that the best way is to try to practice more, just like speaking. I think it's very advisable and I have just started here. Also, I really want to get some other recommendations from you, should you have any advise advice [2] in improving writing, please kindly tell me. I would really appreciate it.

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[1] Here, the verb 'pay' means 'give'.

    "I will pay him a compliment."

    "I will give him a compliment."

With the word 'give', the preposition is 'to'.

    "I give compliments to people."

Because the verb 'pay' means 'give', it uses the same preposition.

    "I pay compliments to people."

This is why I have changed the preposition in your sentence.

[2] The word 'advise' is a verb:

    "I advise you to learn English."

The word 'advice' is a noun:

    "I will give you advice."

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You do not have a "big grammar problem".  You may think that native English speakers write with 100% accuracy—that they never make mistakes.  This is not the case.  In fact, your writing is better than many native English speakers.  Your mistake about 'advise' and 'advice' is a very common mistake with native English speakers.  Please don't be afraid to make mistakes, because native speakers also make mistakes.