That assumes that the starts are on consecutive lines which may be a perfectly valid assumption. It pays to know your data.

Another approach is to use the original regex, which may have multiple starts and then trim it using s/^.*start/start/is.

The loop then looks something like

while ( $string =~ /(start.+?end)/gis ) {
    my $data = $1;
    $data =~ s/^.*start/start/is;
    print $data, "\n\n";
}
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If the intent is to strip off multiple starts only on consecutive lines then the regex would be s/^(?:start\s*)+start/start/is which used on the input

start
start
start
go
one
end
start
start
data
start
go
two
end
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would produce

start
go
one
end

start
data
start
go
two
end
[download]

But as I said you really need to know your data and other factors such as if you need to validate the input.