Can I get admission into the B.Tech CSE program at Thapar University with 19 percentile in JEE? I have 95% marks in Class 12.

Roughly speaking, the total tuition fee for B.Tech at Thapar University comes to around ₹12–14 lakhs for the full 4 years (varies by branch and year).
Hostel and mess are extra.

hapar University offers a wide range of programs. The main ones are:

Undergraduate: B.Tech (multiple engineering branches), B.Sc, BBA
Postgraduate: M.Tech, MBA, MSc, MA
Doctoral: PhD across engineering, sciences, management, and humanities

It's best known for its engineering and management courses.

Thapar University conducts its admission on the basis of both merit and entrance exam scores. For admission in the BTech course candidates must appear for the JEE Mains exam and for MBA they must appear for CAT/XAT/NMAT or the TMAT entrance exam. The application process is online. Students must fulfil the eligibility criteria as per the course.

Thapar University offers BTech course to students at the UG level. It is offered for 4 years in several specialisations. To be eligible for BTech at TIET, candidates must fulfil the eligibility criteria set by the college. Candidates must be Class 12 passed with a 50% aggregate from a recognised board. 

Thapar Institute of Engineering and Technology (TIET) is a private (deemed-to-be) university, not a government institute.