Few reasons:

1. The system is far from perfect. It's not as good as say, the NHS or Canada's health system. And while it's "free" healthcare that is better than the non-existent free healthcare that doesn't exist for other Americans, it's underfunded and understaffed, especially following 20 years of war which obviously saw a huge strain put on the VA system.

2. It's only healthcare. Veterans with untreated psychiatric problems also often struggle with homelessness and stable employment. If they're transient, it can be difficult to insure they, say, make a key appointment to get a diagnosis or prescription.

3. Many people who are largely on their own with psychological issues, including but not limited to veterans, simply do not stick with a treatment regimen. There aren't a lot of mechanisms in place to force someone to take a prescribed drug, even if it helps, and don't like how it makes them feel. This obviously can feed back into #2.

4. Selection bias. It seems like "so many" because our military is huge. 1.9 million US troops were deployed to Iraq and Afghanistan over the past 20 years. Of those let's say 5% saw actual combat (hard to say how many, but all estimates I have seen say certainly not more than 10%), which is 95,000. If most of those end up with PTSD, that's more soldiers than most of the coalition forces sent over, combined. If around half do (and around 35,000 US troops were injured, so this tracks), that's more than France's entire contribution to the invasion and occupation. The vast majority of the remaining 1.8 million who went over and weren't in combat are typically fine. Sure, some will also have psychological issues, but these are people who might have anyway even if they weren't in the military.