IIEST Shibpur - Indian Institute of Engineering Science and Technology

Hello Ship,
No, you cannot take admission in any IIT's or IIEST via WBJEE, via WBJEE you can get state funded government colleges of West Bengal like Jadavpur university, Kalyani, etc.

From 2017, along with 2016 batch, have an option of changing 4-year B.Tech course to 5-year dual degree at the end of the 3rd-year which is called exit option.
So, it should be well cleared and boldly marked that IIEST doesn't offer 5-years as a compulsory course, it's an option at the end of 3rd- year.

IIESTs are a new breed of institutes created by upgrading existing engineering colleges or universities under the various state universities.
Altogether, 4 IIESTs were proposed, IIEST-Shibpur being the first of them.
Before the upgradation too, it is a good institute with healthy teaching and now with the tag of 'Insitute of National Importance' that all NITs enjoy, it's surely a good choice.
Do consider SPAs (School of Planning and Architecture) and also few top-ranked NITs like NIT-Trichy, NIT-Jaipur, etc.

Following space crunch on the campus, IIEST (Indian Institute of Engineering Science and Technology), Shibpur has decided to scrap its five-year integrated dual-degree(B.Tech and M.Tech) programme from the academic session 2017-18. Discontinuing the programme, the institute will bring back its BTech programme after three years. Till date, the institute has admitted around 650 students - every year - to its dual degree programme.

Jadavpur University accepts JEE Mains paper 2 rank. Previous year's cutoff for Jadavpur University was around 11k for general category students.