What you have is close, but you need to:

• add the character selectors . to be quantified (. can work to match everything)
• constrain the end of the string to prevent the length from going over the max with $
• can remove g flag since you only want one match

That leaves you with:

/^\+(?=20).{1,5}$|^\+(?!20).{1,7}$/

which you can also write as

/^\+((?=20).{1,5}|(?!20).{1,7})$/

In your regular expression, (?=20){1,5} reads, "execute the positive lookahead (?=20){1,5} between 1 and 5 times". That has the same effect as executing it once, clearly not what you want. What you want for that part of the expression is

(?=20).{1,5}

which reads, "match between 1 and 5 characters, the first two of which must be '20'".

That's still a problem however, because that matches '+20012' in the string '+20012 quick brown dogs' You need an end-of-string anchor:

^\+(?=20).{1,4}$|...

Notice I also changed {1,5} to {1,4}, as the string may contain at most 5 characters, or at most 4 after '+'. Also observe that the above is the same as

^\+20.{1,2}$|...

which is simpler and arguably reads better. So you could modify your entire regular expression to be

^\+20.{1,2}$|^\+(?!20).{1,6}$/g

or, to reduce repetition,

^\+(?:20.{1,2}|(?!20).{1,6})$/g

For one, that matches '+200X'. It should, unless by,"...when it starts with +20..." you mean, "...when it starts with +20 not followed by a digit...". I have assumed you meant the latter in my solution that follows.

You could attempt to match the following regular expression:

^\+(?:20(\D.?)?|(?!20(?!\d)).{0,6})$

Demo

The expression can be broken down as follows.

^           # match beginning of string
\+          # match '+'
(?:         # begin non-capture group
  20        # match '20'
  (?:\D.?)? # optionally match a non-digit that is optionally
            # followed by any character
|           # or
  (?!       # begin negative lookahead
    20      # match '20'
    (?!\d)  # negative lookahead asserts '20' is not followed by a digit
  )         # end negative lookahead
  .{0,6}    # match between 0 and 6 characters
)           # end non-capture group
$           # match end of string

Note that '+200' is matched by the second part of the alternation (the string does not begin with '20').