I get exactly what you mean
Code
#!/usr/bin/env gnuplot
# Shittiness , Age
$DATA << EOD
89 16
50 20
35 25
15 60
EOD
f(x) = a * exp(-k * x) + c
a = 100
k = 0.05
c = 10
fit f(x) $DATA using 1:2 via a, k, c
set terminal pngcairo size 500,325 enhanced font 'Arial,12'
set output 'age_vs_shittiness_exponential.png'
set title 'Age vs Shittiness — Exponential Fit'
set xlabel 'Shittiness'
set ylabel 'Age'
unset key
unset grid
set xrange [0:100]
set yrange [0:70]
set grid
set key top right
fit_label = sprintf('Fit: %.2f e^{-%.4f x} + %.2f', a, k, c)
plot $DATA using 1:2 with points pt 7 ps 1.6 lc rgb '#cc2222' title 'Data', \
f(x) with lines lw 2.5 lc rgb '#2266cc' title fit_label

but it still doesn't describe the positive trend you're angling for unless you invert the x-axis, which is not how the original image is framed