Let’s start with k=10^3.
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k=10^3;g=1;d=1;{ufun,vfun}=NDSolveValue[{u''[x]-k*u[x]/(g+u[x])==0,v''[x]+k*u[x]/(g+u[x])/d==0,(*u'[0]==0,*)u[1]==1,v[0]==0,v[1]==0},{u,v},{x,0,1},Method->"FiniteElement"];Plot[{ufun[x],vfun[x]},{x,0,1}]
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Now, in order to get to k=10^4. We use a graded finite element mesh
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Needs["NDSolve`FEM`"]
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mesh=ToGradedMesh[Line[{{0},{1}}],<|"Alignment"->"Right","ElementCount"->100|>]
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ElementMesh[{{0.,1.}},{LineElement[<100>]}]
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k=10^4;g=1;d=1;{ufun,vfun}=NDSolveValue[{u''[x]-k*u[x]/(g+u[x])==0,v''[x]+k*u[x]/(g+u[x])/d==0,(*u'[0]==0,*)u[1]==1,v[0]==0,v[1]==0},{u,v},{x}∈mesh,Method->"FiniteElement"];Plot[{ufun[x],vfun[x]},{x,0,1}]
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As a next step you could try to use that solution as an initial seed for a high precision solution.