Turing recognizable but not Turing decidable language cannot have TM do not halt on infinitely many inputs

Question

Sorry, I think I misunderstand the question, It should read as if $L$ is turing-recognizable but not decidable, then there exists infinitely many input that any TM will not halt on it...

Answer 1

The question shows several misconceptions. I'll try to clarify the key aspects.

If $L$ is recognizable but not decidable, then $L$ has to be infinite (otherwise, it is decidable). If a TM recognizes such $L$, it has to accept all the words in $L$ (by definition of "recognizes"), hence it must halt in infinitely many cases.

A TM halting on infinitely many cases does not imply that the recognized language is decidable. (I'm unsure about why you think this would be the case, it would make the halting problem decidable, for instance.)

There is no such a thing as an uncountable language, at least in a conventional setting where the alphabet is countable, and words have finite length. Hence, a TM can never halt on "uncountably many" words.