public void RecordLink()
{
    //Обнуляем текстовый файл
    StreamWriter sw = new StreamWriter(@"text.txt", false);
    int counterNewsLink = 1;

    while (counterNewsLink <= 19)
    {
        counterNewsLink += 2;

        sw.WriteLine(pageLink.Substring(counterNewsLink + 1));
        sw.Close();
    }
}

StreamWriter sw = new StreamWriter(@"text.txt", false);

public void RecordLink()
{
    using (StreamWriter sw = new StreamWriter(@"text.txt", false))
    {
        int counterNewsLink = 1;
        while (counterNewsLink <= 19)
        {
            counterNewsLink += 2;
            sw.WriteLine(pageLink.Substring(counterNewsLink + 1));
        }
    }
}

sw.Close();

using (StreamWriter sw = new StreamWriter(@"text.txt", false))
  for(int counterNewsLink = 3; counterNewsLink <= 21; counterNewsLink += 2)
    sw.WriteLine(pageLink.Substring(counterNewsLink + 1));