You can find out the average transaction fees gathered per block here. You can see that it's been slowly increasing recently, and there tends to be a block every year or so that skews the average quite a bit. But we can confidently say that currently, the average fees per block is around 0.35 BTC, which has increased from about 0.15 BTC of a year ago. Assuming a linear 0.2 BTC increase in fees per block per year, and a 4 year halving schedule we get these equations:
fees = 0.2 * x + 0.35
reward = 0.5^(x/4) * 12.5
Reward is a discrete function, so we can just see what happens at the halving in 2020, 2024 and so on.
fees = 0.2 * 4 + 0.35 = 1.15
reward = 0.5 * 12.5 = 6.25
fees = 0.2 * 8 + 0.35 = 1.95
reward = 0.5^2 * 12.5 = 3.125
fees = 0.2 * 12 + 0.35 = 2.75
reward = 0.5^3 * 12.5 = 1.5625
If it proceeds like this, it looks like there actually won't be a moment when average fees and reward actually equal each other as the reward will halve and will be bypassed by fees at that point suddenly.
A different (linear or non-linear) function for fees will obviously achieve different results as would the block halving time being a bit shorter than 4 years, but this projection does make it seem as though 10-12 years is around when fees and reward will approximately meet.