What is weird about Nick's drive into New York with Gatsby in The Great Gatsby?

While Nick and Gatsby drive into the city, Jay tells about his past, almost as if he is trying to win Nick’s approval. Gatsby is aware of the rumors about him, and he says he doesn’t want Nick to get the wrong impression of him, saying, “I’ll tell you God’s truth.” However, Nick notices some inconsistencies in Gatsby’s story. For instance, Jay claims to have grown up in the Midwest, and when Nick asks specifically where, he replies “San Francisco.” Clearly, Gatsby is unfamiliar with a map of the United States. Also, Gatsby says he was “educated at Oxford” in such a hurried manner that Nick begins to doubt his sincerity. He also claims to have no living relatives at all, which seems odd. Although some of Jay’s story seems suspicions, he is able to produce a photograph and a medal to substantiate his claim that he served in the war. Nick decides to trust him again.
Bringing back an air of mystery, Gatsby tells Nick he is going to ask a big favor of him, but he refuses to talk about it. Nick is to learn about the favor later on from Jordan. Jay says it’s important that Nick know something about him before he asks for Nick’s help.
Nick sees Gatsby’s authority when they are pulled over by a police officer. Jay shows the officer a card from the Police Commissioner, and the officer apologizes for not recognizing him. This incident establishes Gatsby’s importance. Nick learns to expect the unexpected where Jay Gatsby is concerned.

Carraway describes his trip into town with Gatsby as "disconcerting." The details he shares about the content of the conversation soon explain why he feels this way. It is evident that Gatsby, behaving "indecisively," is oddly nervous. As an opening gambit, he asks Carraway "What's your opinion of me, anyway?" which is the sort of frighteningly direct question not generally asked in polite society. It oversteps all the natural boundaries between casual acquaintances.
Gatsby clarifies that he doesn't want Carraway to get "a wrong opinion" of him, believing himself to be much gossiped about. In order to pre-empt these wrong opinions, he declares that he will tell Carraway "God's truth." Unfortunately, his behavior makes Carraway feel he may be lying as he explains that he was educated at Oxford—possibly an explanation for the British affectations in his speech ("old sport"). He continues telling his life story unprompted, its details so "threadbare" that Carraway has to contain his laughter. It is clear that Gatsby is completely unable to read social cues in this situation, as he does not realize how discomfiting this is for Carraway.
Gatsby even presents Carraway with some items he "always carries," including a photograph of himself from Oxford, but Carraway is left with the impression that these items are awkward props. In trying so hard and so ineptly to convince Carraway of "the truth," Gatsby has only succeeded in raising suspicions.

To answer this question, take a look at chapter 4. During their drive to New York, Gatsby says some weird things and acts in a strange way.
First, without any prompting, Gatsby says he wants to tell Nick his family history because he doesn't want Nick to have the wrong idea about him. For Nick, this is a sign that Gatsby is aware of the "bizarre accusations" that people have leveled against him.
Secondly, when Gatsby says that he was "educated at Oxford," Nick notes that he says the words in a very strange manner, as though he has "swallowed" or "choked" on them.
Thirdly, when Gatsby describes how he lived as a "young rajah" in Europe, Nick almost bursts into laughter because Gatsby's description is "threadbare." In other words, it lacks any sense of authenticity, as though he is describing something that never actually happened.
Finally, Gatsby also shows Nick some souvenirs from his past. There is a medal from the war and a photograph from Oxford. It is weird that Gatsby would show these to Nick because it suggests that Gatsby is desperate to prove that his stories of the past are authentic and genuine.