I scored 96 percentile in Jan, & 93% in my 12th boards. SO should I wait for the JEE to take place or shall I just get admitted to private colleges like Thapar college?

Hello,
You can apply for the application form for various private institutions for admission. This year due to current pandemic, many Institutes have cancelled their entrance exams and are taking admission on the basis of merits in class 12. Institutes like VIT Vellore, TIET and many more are taking admission through class 12 marks. Also, students who have qualified for JEE Mains are getting higher preference.

So you can apply for the application form. The last date to apply for VITEEE and TIET is July 31 2020. Also please keep on preparation for your JEE Mains second session. You can get seats in better NITs, IIITs and GFTIs.
Even with your current percentile, you can get some good branches in the newer NITs and IIITs. Your rank would be around 35k in the common rank list.  So please kepo on practicing and try to improve your score in the second session of JEE mains which will be conducted from September 1st to 6th, 2020.

You can enroll for the JEE Mains crash course by Careers 360 which provides video lessons frequent mock tests and previous year question papers and many more for just Rs 6999. Please visit this link https://www.google.com/url?sa=t&source=web&rct=j&url=https://engineering.careers360.com/articles/careers360-jee-main-crash-course/amp&ved=2ahUKEwiq8sH4iPDqAhUFT30KHW-KBHAQFjAAegQIAxAB&usg=AOvVaw2qqSyBCop2N7ziMSe61Y3S&ampcf=1

Hope it helps