The military offered condolences to the families of the martyred soldiers and wished injured soldiers a speedy recovery.
Turkey launched Operation Olive Branch on January 20 to clear terrorist groups from Afrin in the northwest of the warn-torn country, amid growing threats from the region.
A British woman fighting alongside Kurdish forces in the Syrian region of Afrin was killed in the Turkish-led offensive on the area, a spokeswoman for the forces said Monday.
Formidable guerrilla fighters, the YPG withdrew from Afrin on Sunday after a Turkish thrust into the town centre.
He added that Turkey will restore peace in Afrin soon.
"Further operations beyond the border regions will draw in more forces and further exacerbate an already deteriorating humanitarian situation", Colonel Rob Manning, a Pentagon spokesman, said in a statement.
Ankara views the PYD, and its military wing, the YPG, and the YPJ, as a "terrorist" group and an extension of the outlawed Kurdistan Workers' Party (PKK), a group fighting a decades-long insurgency against the Turkish government in Turkey. "If you can't handle it, then we may suddenly enter Sinjar one night and clear out the PKKs there". Relations with the USA have been strained over Washington's support for the YPG, and a direct confrontation between their forces is liable to ignite over Turkey's vow to clear Kurdish fighters from areas where US forces are embedded.
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"The final destination is a Turkish war on the Kurds throughout northern and eastern Syria that destroys the YPG and forces the Americans to work by, with and through Turkey" in the fight against IS and in Syria, he said.
United States support for the Kurdish-dominated SDF has infuriated Turkey and created an ongoing diplomatic crisis between the two North Atlantic Treaty Organisation allies.
"A YPG insurgency in Afrin would prompt Turkey to escalate against the Kurds throughout northern Syria, jeopardizing the US -led mission to stabilize the areas captured" from IS militants, he said.
Many Kurdish fighters left the fight against Daesh to defend Afrin and the BBC says some British volunteers are known to have joined them.
Images emerged of torched stores and men, some in uniform and others in civilian clothes, walking out of homes with household goods.
The US State Department also expressed concern over the humanitarian ramifications of the Turkish operation in Afrin.
The situation in Afrin has seen some Kurdish members of the SDF abandon the battle against IS in the Middle Euphrates River Valley and head west to join the fight against Turkey and Turkish aligned forces.