How realistic is the science behind the Lightyear movie? Buzz Lightyear flies into space, and accelerates up to 70% of the speed of light. When he lands, suddenly four years have past back home, while it was only about 5-10 minutes for him. Is that realistic?
It makes no sense for several reasons.
The gamma factor for $0.7c$ is only $1.4$, so a few minutes for him would just be a few more minutes for those who stayed behind.
The flight plan slingshots around the star Alpha Takoni and starts and ends on Takoni Prime, which is a class M planet that presumably orbits Alpha Takoni, making it only a few light minutes away. That means a roundtrip to the star at anything close to $c$ would last only a few minutes for those who stayed behind. As his speed approached $c$, the $γ$ factor would approach infinity, but that's the factor by which the trip is shorter for him. The only way the times would work out is if the star was actually light years away and his speed was $0.99999...\,c$.
In the second clip, Buzz says "the faster I fly, the further into the future I travel", as though repeating the trip at a higher speed would take even longer. That's not how speed works.
They make it clear that he can't decelerate to turn around at the star—he has to slingshot. You can't slingshot around a star at a substantial fraction of the speed of light. The amount your path will deflect at that speed is a scarcely measurable fraction of a degree, not a full 180 degrees.
(I have not watched the movie and most of the following answer was written for the question before it was edited.)
No. 70% of the speed of light results in a Lorentz factor of only 1.4, which means that 1 minute of rocket time corresponds to 1.4 minutes of Earth time. That is not nearly enough to produce time dilation of the magnitude you have in your question.
In lightyear they completely and utterly botch how relativity works. They clearly heard someone say "time slows down the faster you go" and took that to mean "the ship moves faster"
To put it simply, time should slow down for Buzz on the Ship, but that doesnt mean the ship actually moves slower.
To put this nonsense into perspective, the faster the ship is going in the movie... essentially, the longer its taking to complete its trip to the people on the planet... thats not how anything in any universe works.
the faster something goes, the faster it goes, there is no universe where moving makes oyu take longer to get somewhere.