Is it historically inaccurate that Jesus is usually portrayed as European looking? Is it historically inaccurate that Jesus is usually portrayed as European looking, having certain facial and bodily features (muscular)? 

Would most people follow Him if he did not look white or was heavy set? It almost seems like a trick sometimes to get certain people to believe.

Does the Bible say anything about his physical appearance and how we are to physically think of Him?



vs.  

(A special on PBS about scientists who put together a large study to determine what Jesus looked most like, and it is not how he was usually pictured.)
With regards to ethnicity and geography, the direct answer to this is a big Yes, it is wrong. Bethlehem is in Palestine, Asia. The borders of this nation is the red sea, Egypt, Lebanon Jordan, the Dead Sea and Syria, (varied with time). The closest picture we could estimate should be that of an Arab or middle eastern.

Theories that Jesus was European were forged, and claims that he was black, partly because Jesus lived as a small boy in Egypt, are inaccurate as well. 


  "...the angel of the Lord appeareth to Joseph in a dream, saying,
  Arise, and take the young child and his mother, and mother and flee
  into Egypt...that it might be fulfilled which was spoken of the Lord
  by the prophet, saying, Out of Egypt have I called my son." (Matthew
  2:13-15).


These claims do not have any  concrete scientific bases and are purely subjective. 

Nowhere in the bible is was Jesus physically described apart from the transfiguration. We are only told of his descent Solomon and David, all from Israel and over 14 generations afterwards.


  "And Jesse begat David the king; and David the king begat Solomon of
  her that had been the wife of Urias; And Solomon begat Roboam; and
  Roboam begat Abia; and Abia begat Asa;....And Jacob begat Joseph the
  husband of Mary, of whom was born Jesus, who is called Christ."
  (Matthew 1:6-16)


The names couldn't be European either.

Another possible answer also by logic is that God could not give him any look different from the people around him. 

Let's assume that Christ was born in Eastern or West Africa where you can find the darkest people on earth. For Him to live an exemplary life so that people wouldn't find it a big deal doing what he did, he must have had the looks of blacks, not white to look strange for his black society to think it takes something supernatural to live a righteous life. 

Remember that the society could not doubt him as the son of Joseph, the carpenter.


  "Is not this the carpenter’s son? is not his mother called Mary? and
  his brethren, James, and Joses, and Simon, and Judas?" (Matthew
  13:55).


Remember Jesus hungered, slept, wept and got angry to prove his earthly being. The best assumption is that, in the Middle-east, he would be born Middle-eastern as some sort of placebo to win friends and followers.