So after when Jenny dies and it is just Forrest and little Forrest we flash forward 20 years and little Forrest just graduated from Harvard Law School. Little Forrest then undergoes all the requirements it takes to be able to run for Governor of Alabama. He wins and then after years in office being a successful Governor he runs for President of the United States. He married his highschool sweetheart Rebecca and they have 2 sons.
Little Forrest had many reason to run for president. One of the reasons is that he saw the toll war took on Lieutenant Dan and how veterans did not have a lot of benefits or real help with PTSD. Lieutenant Dan was convicted of murdering his wife. He had a war flashback and thought he was in Vietnam again and strangled his wife. That was one of the big things he proposed, to help soldiers when they come home.
Next thing he advocated was better research on HIV/AIDS. Jenny as we all know died of AIDS, even though the movie never said it, it was heavily implied because at the time the no one new what it was. Soon after later his father Forrest Senior got diagnosed with AIDS that he contracted from Jenny. While he was Governor he had marches and fundraisers to get research facilities open in Greenbow Alabama to save his dad. Forrest Sn. is still cutting grass and playing ping pong. But he got a lot of backlash with this because HIV/AIDS was most time correlated with the gay community. Little Forrest was even accused for being gay even when he had a wife. His sons were accused as well.
But after that his shifted more fucus into helping people with learning disabilities. He knew his father had a hard time with school because of his disability. He saw it first hand when his son Thomas was diagnosed with autism and school was difficult. So he opened up after school programs to help children.
He became President and did all he said. He found a cure for AIDS, help veterans get back on there feet and the the drop out level in high school dropped significantly.