The speaker speakoke rapidly.

I almost didn’t understand him.

But, I realized his English may be British English.

He said “perhaps “."

I still can’t catch the difference of English accents only from listening.

But, I can immediately understand standard Japanese or Kansai dialects.

Anyway I want to usespeak English fluently.

This morning, I listened to a TED talk.

The speaker speakoke rapidly.

You could combine these two sentences. The speaker from the TED talk I was listening to this morning spoke rapidly.

I almost didn’'t always understand. him.

I struggled to understand him. OR I didn't always understand. Your sentence structure is correct but it just didn't sound right to me in the context. Missing a subject maybe which is why I added "him".

But I realized his English maybe British.

In general, avoid starting sentences with "but". Again, you could combine these two sentences to make more complex sentences. "I struggled to understand him but I realized it is maybe because he was using British English."

He saiused the word “perhaps “.

Nothing really wrong with the sentence, just could make it more complicated.

THe said “perhaps “, this word is often used British English, right?

I, again, think you could combine these two sentences.

I still can’t catchhear the difference ofbetween English accents only listening.

Not wrong entirely, just another way to say it. "I still can’t catch the difference between English accents when listening to them.

ButHowever, I can immediately understand standard Japanese or Kansai dialects.

When writing, the avoiding "But" at the start of the sentence is important. Not as important when speaking.

Anyway , I want to usespeak English fluently.

The speaker speakoke rapidly.

Youre talking about something that happened in the past, so use past tense

I almost didn’t understand him.

Better to make “understand” a transitive verb

But I realized his English is maybe British.

This word is often used British English, right?

Also in formal writing too!

I still can’t catch the difference ofs in English accents by only listening.

The speaker speakoke rapidly.

I almost didn’tThis is perfectly correct grammatically, but it sounds better if you say something like "It was hard for me to understand".

But I realized his English may be British.

Maybe = adverb that means "perhaps"
May be = verbal phrase that means "might be"

In this case, an even better word would be "could" : I realised this English could be British"

I still can’t catch the difference ofbetween English accents only by listening.

Anyway, I want to use English fluently.