2 columns Date (type date) (empty) and Date_text (varchar) full of dd/mm/yyy text date.
That's British date format and depending on the default language setting of you SQL login the engine expects more US format.
But you can pass the format information to CONVERT function for conversion, here British = 103, see CONVERT => Date and Time styles
Additional option is to use the TRY_CONVERT function, which don't raise error, it returns NULL for not-convertable values instead.
So try it with
UPDATE DATE_TABLE
SET Date = CONVERT(date, Date_text, 103);
UPDATE DATE_TABLE
SET Date = TRY_CONVERT(date, Date_text, 103);