The bigger question is really: where would it go?
Because of course matter isn't created or destroyed. So it'd have to go elsewhere.
The only two places it could go are:
- Into the Earth. But water is fairly light (1 g/mL versus Earth which appears to increase from 1.02 g/mL near the surface to 13.09 g/mL at the center). Plus, even if water could/did continue to transition into the Earth in large amounts, it'd just build up there until it reached a saturation level. And there's only so much room between the highly compressed rock.
- Out to space. However gravity generally keeps the atmosphere here, just like it does for us. A very small percentage of gas does escape. However, it's predominantly lower mass gases like hydrogen and helium. The question on our site of What “g” would be needed to keep helium on Earth? is quite useful for information on this.
Or, alternatively,
- Out to space as its constituent gases. (Monotonic) Oxygen and especially Hydrogen do escape somewhat more readily. Electrolysis and other water splitting methods do cause a small percentage of Earth's water to be broken down at any given time. However, based upon this estimate of 95,000 tons of hydrogen lost per year, and this estimate of $1.4 \times 10^{21}$ kg of water, it'd take 1.8 trillion years to lose all of Earth's water by hydrogen loss.
Otherwise, if it isn't leaving the system, it could only build up in the atmosphere... potentially until reaching a level where the air could hold no more. Theoretically that'd be at 100% relative humidity. Unfortunately the troposphere can only hold the equivalent of a layer of a couple inches\few cm of water. Additionally water tends to leaves the air when enough areas of it reach 100% RH, through processes like dew and precipitation.
And so over the long run, it can only reach an equilibrium, where the amount of water the air held was in balance with the amount it lost. And that seems in line with what we see. Moisture levels vary, but aren't rapidly changing in any direction. Considering that there's 96000 times more water in the oceans than there is in the air currently, the atmosphere surely couldn't hold it all anyways! If it can't leave, and the atmosphere cannot hold it all... well, it has to stay on the Earth. Certainly we see the equilibrium in practice too, as sea\lake levels aren't dropping consistently. Lake levels actually vary quite a bit, showing evaporation has the ability to drain at least the lakes fairly fast. Yet they aren't dropping consistently in any noticeable way. We're very dry in Florida right now, and lakes are way down... but in places like California and the Piedmont they've been seeing great flooding. But neither lasts in the long run.
Looks like studies do estimate that the Earth has lost 1/4 of its water in its estimated lifetime. So indeed it's not that nothing leaves. It's just that it's very slow. If it wasn't... well we'd be seeing it quickly change now! Over our lifetimes, over the past few thousand years. But there's no indications that general lake/ocean levels have drastically changed.
I do really get that there can be skepticism at billions of years. It's certainly something we can't verify too easily, visually, personally. And with Biblical\other religious sources casting potential dispute upon it, that increases the question.
Honestly, a reasonable scientist should welcome skepticism and try to address it with fairness and levelheadedness. Unfortunately there certainly are plenty who are dismissive and belittling, either openly, or in their attitudes.
But while questioning is a very worthwhile endeavor, this particular line of reasoning doesn't appear to hold much mustard. Be wary not to jump too quickly on any potential disagreements you see to accepted theories and thus run hard with them, or you're no different from those overly rigid scientists. Keep seeking for the truth as something of the greatest value. Regardless of what it is, and in the end it alone shall stand, and shall set you free!