TNT field solver - howto handle 'air' between ground plane and wire?
A rough rule of thumb is that a 50 Ohm microstrip will be close to the same width as its distance to the reference plane. So your simulation isn't that surprising.

You can reduce the microstrip width by choosing a thinner board, or using top copper fills as you show in your question. The copper needs to be close to the microstrip to have an effect. And it needs to be well stitched to ground. Google "coplanar waveguide".

A 4-layer board will give you a much closer reference plane :)

As far as TNT, try this: to your cross-section, add "New Rectangle Conductors" and name them "groundWires" (the capitalization is intentional, but don't use the quotes). Then, in the details, choose number=2 and adjust the spacing accordingly. The spacing won't be intuitive :)