The best indie games like OutRun are Horizon Chase 2, Slipstream and Art of Rally. Below are 5, each with the reason it is here — and, at the end, what filtering to indie costs you.
Filtering to indie means you want the answer a small team would give. It is usually the most interesting filter on this site, because small studios take the risks large ones cannot.
What you get: One strong idea, explored completely, at a price that makes trying it easy. Indies have no production to hide behind, so the design is exposed.
What made OutRun worth playing in the first place is covered on the full list. This page is the indie part of that answer, and only the indie part.
Made explicitly to be arcade racers of this era again.
A pixel-art love letter to OutRun and Lotus Turbo Challenge.
Driving as an aesthetic experience rather than a competition.
Low-poly arcade racing with the same clean feel.
For the open-road tone taken somewhere else entirely.
Polish and scale, sometimes. A few of these are rough at the edges, and a big-budget pick will always look better doing the same thing.
Specifically, these are strong answers for OutRun that are not indie games, so they are not on this page:
The full list is on games like OutRun — 9 picks, no genre filter.
Horizon Chase 2. Made explicitly to be arcade racers of this era again.
No. OutRun released on the Sega Genesis, but a recommendation is only useful if you can play it, so the picks above span platforms and eras. What they share is the genre and the reason.
Self-published, or published without a large publisher's money. It says nothing about the size of the game, which is why a forty-hour metroidvania and a two-hour art piece sit under the same label.
Constraint. Without a budget to hide behind, one idea has to carry the whole thing — and a small team can commit to something risky that a large one would sand down.
By hand. Each pick was written for the full OutRun list first, on what the game actually does, and this page is the part of that list that is indie — never a genre tag lookup, which is how "best racing games like Super Mario 64" used to be a page on this site.
Or read the indie guide for where to start in the genre generally.