More drainage issues w/ picture. Need advice.
Different configuration, etc., but I had a similar situation, for which I had considered backfilling, and decided that may only move the problem to another area of the lawn, so we ultimately cut what for all intent and purpose is a ditch in part of the yard this spring. The low area, which we referred to as a pond every spring and after heavy rains, was abour 4-6" below the surrounding terrain, but further out it dropped off toward the swamp, similar to how yours drops toward the street. So, I dozed a swath about 8' wide with the tractor, after which we backfilled, graded, and raked the edges, to create a very gradual low area of its own, so that it blends well with our natural rolling terrain, and allows a path for the water to travel. I just seeded it last week, so it's not completely done yet, but it has already shown to be working as planned. Another option that I had considered, depending on how much fall you have in front, may be to install a catch basin, install a buried tile, and route it to daylight. With either of these though, you could have a compliance issue with your local municipality, by creating unnatural flow to the city sewer.