The closest library to you probably has Consumer Reports. The April issue every year is devoted to rating automobiles, including recommended used cars. $5,000-10,000 is the lowest price range of recommended used cars.
Some makes/models and year of manufacture (link below):
with taxes, registration, insurance, maintenance and repair, that 4k isn't going too far. that said, get a well maintained toyota corolla with a dent or two for 2k. get new tires, oil changes, brakes and coolant. make sure ac blows ice cold.
something like this can work: https://losangeles.craigslist.org/lac/for/d/universal-city-2001-corolla-sport/6877089729.html
camry or accords are too much money and more difficult to park
here's a list from autobytel:
https://www.autobytel.com/car-buying-guides/features/10-best-used-cars-under-3-000-121519/
One of my employees spent the last 15 months going through 4 different used cars because he trusted “a friend” to hook him up deals. And he regretted it within weeks. Over and over again.
You do not have the time or luxury to go over repairs. So spend the $$ and get something professionally inspected.