Creating a regex expression for target URL

That isn't even close.
To select "http: or https:" you would need teh Regex "OR" operator '|'

^((http:)|(https:)).*$

Matches any string that starts with "http:" or "https:"

But even with that, your rules are somewhat arbitrary: a valid URL can be much longer than 63 characters, and can include a much wider range of characters. ANd they don't have start with "http" at all - the browser will happily infer that.

Personally, I like regexes - but they can quickly become unwieldy and complicated, which makes them prone to error and hard to fix. I'd probably do the validation in C# in a couple of goes:
1) Does it start with "http:" or "https:"
2) Is it the right length?
...
That way, you get much more readable code that is a lot easier to modify when one of your rules is found to be wrong ... which they almost certainly will be!