The real founder of Turkish rule in Bihar and Bengal, who sacked Nalanda and Vikramashila around 1200 CE, was:

Ikhtiyaruddin Muhammad bin Bakhtiyar Khalji, a Turkish (Khalji) commander of Muhammad Ghori and Qutbuddin Aibak, is regarded as the real founder of Turkish rule in Bihar and Bengal.

About Bakhtiyar Khalji:

• He was rejected for service in the armies at Ghazni and Delhi because of his poor appearance, according to tradition, but later found service in Awadh.
• He began raiding the territories of Bihar from his base in eastern Uttar Pradesh.

His conquests:

• Around 1200 CE, he attacked Bihar and captured the fortified town of Odantapuri (Bihar Sharif), mistaking the monastery there for a fort.
• He sacked the great Buddhist monastic universities of Nalanda, Vikramashila and Odantapuri, killing many monks and burning their libraries.
• This was a severe blow to Buddhism in India.
• Around 1204-05, he made a sudden attack on Nadia (Nabadwip), the capital of the Sena king Lakshmana Sena, with only a small band of horsemen; the old king fled.
• He then made Lakhnauti (Gaur) his capital, and established Turkish rule in north and west Bengal.

His end:

• In 1206, he led an expedition to Tibet through Assam, which ended in disaster; most of his army perished.
• He returned to Devkot, where he died (or was murdered by one of his officers, Ali Mardan Khalji) in 1206.

Source:

The main source for his campaigns is the "Tabaqat-i-Nasiri" of Minhaj-us-Siraj, who visited Bengal later.

After him:

• His successors, such as Ghiyasuddin Iwaz Khalji, ruled Bengal and sometimes defied Delhi.
• Iltutmish brought Bengal under Delhi's control.
• Bengal later became independent under the Ilyas Shahi and other dynasties.

Why not the other options?

• ✗Qutbuddin Aibak and Iltutmish ruled from Delhi.
• ✗Ghiyasuddin Iwaz Khalji was a later ruler of Bengal.