This might be hard to answer according to /r/AskHistorians standards, since Jesus certainly wasn't depicted as white "throughout history". Absolutely he was in European / Euro-American art, from which you might recognise some very famous depictions: Christ Pantocrator (Orthodox: Hagia Sophia, Byzantine Empire, 13th century), The Last Supper (Catholic: Leonardo Da Vinci, Italy, 14th century), Head of Christ (Evangelical Protestant: Walter Sallman, United States, 1948) etc. But despite their massive fame, these certainly aren't the only depictions of Jesus around, and plenty of people - Christians and non-Christians alike - from outside Europe and North America have had a crack at it.

Ethiopian Orthodox Christianity, for example, has developed as the state religion of Ethiopia since the fourth century, independent of the European tradition (due partially to theological reasons, but also simple geography) and their depictions of Jesus often appear distinctly Ethiopian (c. 18th century) - no wavy blonde locks here. Among the small Chinese Christian minority, too, Jesus takes on an East Asian appearance (19th century). All this goes to show is that artists and devotees were inclined to portray Christ with the same features and style as they saw day-to-day, and could identify with most easily. Hence, naturally, the European and modern American depictions as well.

With the advent of 18th/19th-century European colonialism, and its introduction of Christianity to regions where Christians were either non-existent or a small minority, European depictions of Jesus often became canonical (so to speak) and native converts generally began to worship him in that style. With modern mass-printing and transmission techniques, this has only intensified - for example, the huge popularity of the aforementioned Head of Christ as shown here (Dambulla Temple, Sri Lanka, 2017 - personal photo). The depiction of Jesus as white, therefore, isn't really "throughout history", but for most regions of the world a relatively recent development.

As to whether people actually believed he was white, I'd have to let someone else answer that one, as I'm definitely not qualified. To get a reasonable answer, though, I think you'll probably have to narrow the place and time period, since people's relations to Jesus, depictions of him, Christianity, and even religion as a whole have changed hugely over the past two millennia pretty much everywhere. Fourth-century Romans, 15th-century Russians, Taiping Chinese, German Nazis, and 21st-century Colombians all had wildly different perspectives (and that's just among self-described Christians!), and I don't think anyone in the world is qualified to speak for all of them. This, I believe, is the reason for Rule 3 on this subreddit.