Certainly, it is not exactly that. But, with the introduction of Buddhism in Japan, many Sanskrit words entered Japanese in a broken form. Buddhist Mantra in Japanese is almost the same as it is in the original form with a little different pronunciation. It is the same also with the Chinese and Korean languages. Many Sanskrit words.apki duaen chaiye mulakks ko
Do not forget the Japanese languages originated from Sanskrit as well.
About the origin of Turkic language, only @Bilal9 can answer, too difficult for me to understand the influence.