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    fit1 = 
Plot[f[t], {t, 0, 65}, 
AspectRatio -> 1, ImageSize -> 200, PlotStyle -> Red, 
AxesLabel -> {Time[fs], (R^2)[\[Mu]m^2]}]

    fit1 = 
Plot[f[t], {t, 0, 65}, 
AspectRatio -> 1, ImageSize -> 200, PlotStyle -> Red, 
AxesLabel -> {Time[fs], (R^2)[μm^2]}]

plot1 = Graphics[Line[data], Axes -> True, ImageSize -> {200}, AspectRatio -> 1, PlotRange -> {{0, 100}, {0, All}}*1]

And then fitting a curve of the form x=a*t^n using

fit1 = Graphics[Plot[f[t], {t, 0, 100}, AspectRatio -> 1, ImageSize -> {400}, AxesLabel -> {Time [fs], (R^2 ) [\[Mu]m^2]}]]

Which gives me two curves and I want to find the intersection points of these graphs fit1 & plot1.

I have tried using GraphIntersection, Solve, and NSolve but none of which have worked because of the data structure (I can write out my implementation for any of these because I may be using them wrong!). I was wondering if there is a way to do this? I can list data if necessary. Thanks for your help in advance!

plot1 = 
ListLinePlot[data,  
Axes -> True, ImageSize -> 200,  
AspectRatio -> 1,  
PlotRange -> {{0, 65}, Automatic}]

And then fitting a curve of the form x = a*t^n using

    fit1 = 
Plot[f[t], {t, 0, 65},  
AspectRatio -> 1, ImageSize -> 200, PlotStyle -> Red, 
AxesLabel -> {Time[fs], (R^2)[\[Mu]m^2]}]

Which gives me two curves and I want to find the intersection points of these graphs

Show[fit1, plot1] 

I have tried using GraphIntersection, Solve, and NSolve but none of which have worked because of the data structure (I can write out my implementation for any of these because I may be using them wrong!). I was wondering if there is a way to do this? The list data is given below. Thanks for your help in advance!

data = {{2.07562, 0.00027459}, {3.01908, 0.000885945}, {3.96254, 0.00188434}, {4.906, 0.00354878}, {6.03816, 0.00730301}, {6.98162, 0.0127167}, {7.92508, 0.0212226}, {9.05723, 0.0342637}, {10.0007, 0.0462345}, {10.9442, 0.0586074}, {12.0763, 0.0760596}, {13.0198, 0.0933376}, {13.9632, 0.11305}, {14.9067, 0.137074}, {16.0388, 0.16735}, {16.9823, 0.192585}, {17.9258, 0.217247}, {19.0579, 0.248665}, {20.0014, 0.277623}, {20.9449, 0.305502}, {22.077, 0.339465}, {23.0205, 0.369906}, {23.9639, 0.401977}, {24.9074, 0.434157}, {26.0395, 0.477391}, {26.983, 0.511391}, {27.9265, 0.542689}, {29.0586, 0.579214}, {30.0021, 0.607107}, {30.9455, 0.639067}, {32.0777, 0.674537}, {33.0212, 0.709506}, {33.9646, 0.744078}, {34.9081, 0.777238}, {36.0402, 0.817115}, {36.9837, 0.85196}, {37.9272, 0.886969}, {39.0593, 0.937822}, {40.0028, 0.979181}, {40.9462, 1.01858}, {42.0784, 1.06353}, {43.0219, 1.09336}, {43.9653, 1.11857}, {44.9088, 1.14213}, {46.0409, 1.16449}, {46.9844, 1.18593}, {47.9279, 1.20096}, {49.06, 1.22164}, {50.0035, 1.23876}, {50.9469, 1.2508}, {52.0791, 1.27807}, {53.0226, 1.30499}, {53.966, 1.3368}, {54.9095, 1.36667}, {56.0416, 1.40522}, {56.9851, 1.43608}, {57.9286, 1.47795}, {59.0607, 1.54707}, {60.0042, 1.6163}}