Minimum tire width to avoid snake bite
I doubt you're getting snake bites in a 22mm tyre unless it's very low on air. Even then it's not likely. 

Snake bites have a distinctive two-hole pattern so you can check. It's more likely that you're getting ordinary punctures. There may be tough 22mm tyres, but most this thin are designed to be light. There are certainly anti-puncture options at 25mm which should fit, but may be a touch slower.