As is often the case, adding "use strict" and "use warnings" gives a clue:

Bareword "old" not allowed while "strict subs" in use at ./re line 7.
Bareword "g" not allowed while "strict subs" in use at ./re line 7.
[download]

B::Deparse is another useful tool.

$ perl -MO=Deparse re
my $one = 'TestnewTest';
$one =~ s//new/;
'???';
'???';
print "$one\n";
re syntax OK
[download]

The delimiters for the substitution operator can only be single characters[1]. Your code is therefore being interpreted as:

$one =~ s//new/; 'old';; 'g';
[download]

So the first empty string Perl finds (at the start of $one) is changed to "new".

[1] I feel sure that someone will be along shortly to point out exceptions to that rule :-)