How to Discretize the Following Region Using Reasonable Bounds?

making that parameter exact seems to fix things:

curve = DiscretizeRegion[
  ImplicitRegion[
   x^2 + x + y^2 + Sin[4*x*y] + Sin[3*x*y] == 
    39/10, {{x, -3, 3}, {y, -3, 3}}]]

enter image description here

note with the exact parameter and your origial "oversize" bounds you get a poor discretisation:

enter image description here

( remedied with MaxElementSize , etc )

Give DiscretizeRegion a second argument specifying bounds:

DiscretizeRegion[
    ImplicitRegion[
        x^2+x+y^2+Sin[4*x*y]+Sin[3*x*y]==3.9,
        {{x,-3,3},{y,-3,3}}
    ],
    {{-3,3},{-3,3}}
]

enter image description here

DiscretizeRegion[
  ImplicitRegion[
  x^2 + x + y^2 + Sin[4 x y] + Sin[3 x y] == 3.9, 
  {{x, -5, 5}, {y, -5, 5}}],
  AccuracyGoal -> 8]

enter image description here

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