An awk solution would be much simpler (and much faster). Simply read all lines containing "Today" into an array in awk. Then beginning with the line not containing "Today" write the current line followed by the associated line from the array, e.g.
awk '/Today/{a[++n] = $0; next} {printf "%s\t%s\n", $0, a[++m]}' file.txt
Example Use/Output
With your example lines in file.txt, you would receive:
$ awk '/Today/{a[++n] = $0; next} {printf "%s\t%s\n", $0, a[++m]}' file.txt
https://123 Today, 12:34
https://456 Today, 21:43
https://789 Today, 12:43
Or if you wanted to change the order:
$ awk '/Today/{a[++n] = $0; next} {printf "%s\t%s\n", a[++m], $0}' file.txt
Today, 12:34 https://123
Today, 21:43 https://456
Today, 12:43 https://789
Addition Per-Comment
If you are receiving whitespace before the output with awk that is due to having whitespace before the first field. To eliminate the whitespace, you can force awk to recalculate each field, removing whitespace simply by setting a field equal to itself, e.g.
awk '{$1 = $1} /Today/{a[++n] = $0; next} {printf "%s\t%s\n", a[++m], $0}' file.txt
By setting the first field equal to itself ($1 = $1), you force awk to recalculate each field which would eliminate leading whitespace. Take for example your data with leading whitespace (each line is preceded by 3-spaces):
Today, 12:34
Today, 21:43
Today, 12:43
https://123
https://456
https://789
Using the updated command gives the answers shown above with the whitespace removed.
Using paste
You can use the paste command as another option along with the wc -l (word count lines) command. Simply determined the number of lines and then use process substitution to output the first 1/2 of the lines followed by the last 1/2 of the lines and combine them with paste, e.g.
$ lc=$(wc -l <file.txt); paste <(head -n $((lc/2)) file.txt) <(tail -n $((lc/2)) file.txt)
Today, 12:34 https://123
Today, 21:43 https://456
Today, 12:43 https://789
(above, lc holds the line-count and then head and tail are used to split the file)
Let me know if you have questions or if this isn't what you were attempting to do.