Ankara sees the YPG as a terrorist group linked to the PKK, which has waged a deadly insurgency in Turkey for over three decades.
The PKK/PYD is the Syrian offshoot of the PKK, which is listed as a terrorist organization by Turkey, the US and EU.
A senior Turkish official said the US's training of the new "Border Security Force" was the reason for Ankara's summoning of Washington's charge d'affaires on Wednesday last week. Rojhat Roj said shelling of areas in Afrin district, in Aleppo province, had killed one YPG fighter and injured two civilians.
Speaking at the provincial congress meetings of his ruling Justice and Development (AKP) in the eastern province of Elazıg on Saturday, Erdogan said "If the terrorists in Afrin do not surrender, we will destroy them".More news: Sanchez-less Arsenal lose at Bournemouth
The Turkish President also attacked Washington saying that "the United States thinks it has established an army of looters in Syria, and it will see how we will destroy these thieves in less than a week,"expressing disappointment towards the U.S. position".
But the few observation posts which the Turkish army says it has established are close to the dividing line between Arab rebel-held land and the Kurdish-controlled region of Afrin.
The Turkish forces were firing from Turkey's southern province Hatay and the observation point in Syria's Idlib province.
In December the Associated Press reported that the USA was developing an expanded training programme for Kurdish and Arab border guards in Syria to prevent the resurgence of Isis.More news: Lebanon blast: Hamas official wounded in vehicle bomb attack
Ibrahim Kalin, the presidential spokesman, said the USA was taking steps to legitimize and solidify the YPG.
"I hope that during an Afrin operation, these powers will not make the mistake of appearing to be on the same side as a terror organisation", Mr Erdogan said in an apparent reference to the USA during a rally in the northern Turkish city of Tokat.
Erdogan said the new operation into Afrin would be an extension of Turkey's 2016 incursion into northern Syria, which aimed to combat IS and stem the advance of US -backed Kurdish forces. Hediye Yusuf, a senior Kurdish official, wrote on Twitter that the Turkish operation against Afrin was a "violation" of the Syrian people and undermined worldwide efforts to reach a political solution in Syria.
Roj said the Kurdish militia will fight to "defend our gains, our territories". More than 40,000 people in Turkey have been killed since the 1980s after the PKK launched its rebellion.More news: California mudslide casualties increase to 18