DC isn't unusually dense. It's only the 6th most dense city in America and not even in the top 100 for the world. It doesn't even have an districts that are in the top 100 (Paris, on the other hand, features several districts and still has good public transport networks).
And this is just looking at density in terms of population. There's also building density, London would rank very high up there and has excellent public transport.
The problem in DC isn't technical; the problem is cultural. You see the car as a solution so you keep investing in roads and under funding public transport. Whereas Europe took a different approach. Some UK cities have "park and ride" schemes where you parks on the outskirts and get a cheap bus into the city. Buses will have their own dedicated lanes too so aren't subject to congestion. Some of these cities even go as far as pedestrianizing chunks of the city center so the only way to access it is via subway, bus, tram or bicycle.
The benefits of improving public transport isn't just reducing congestion either. You improve the transport for the vulnerable (elderly, poor, etc), you improve the air quality in the city, you improve road safety. It's better for the environment, it's better for peoples health, it's better for moving people around. But it requires a cultural shift to happen.