Your last screenshot looks like it's probably right.

BTW, these two are the same thing: vertex_output.tbn = transpose(mat3( tangent.x, bitangent.x, normal.x,
tangent.y, bitangent.y, normal.y,
tangent.z, bitangent.z, normal.z ));//manually transpose, then (un)transpose

vertex_output.tbn = mat3(tangent, bitangent, normal);

vertex_output.tbn = mat3(tangent, bitangent, normal);
Should you renormalize that at fragment shader? Now values are linearly interpolated but how valid the transformation is? If you send unnormalized tangent, bitanget and normal and then normalize those at fragment shader and create matrix there you might get better results. But havent tested it myself so I can't say is there any real difference.

Interpolated normals does not stay normalized. This is basic thing that is done for every lightmodel. Normal is renormalized at pixel shader.

Try to lerp between [1,0,0] and [0,1,0] and you clearly see the unormalized results.
But problem is more like. Is it needed for normal mapping? Or is the problem too small to be notified.