Wright was the third son of John Wright of Wright's Bridge, near Hornchurch in Essex, was born in 1590.He matriculated a pensioner of Emmanuel College, Cambridge, in March 1608, and proceeded B.A. the following year.
He was admitted a candidate of the Royal College of Physicians on 22 December 1618, elected fellow on 22 December 1622, censor in 1628 and 1639, named an elect on 24 May 1642, conciliarius in 1647, and again from 1650 annually till his death in 1657.