Do supports have enough health?

I was just playing some nami{{champion:267}} with trist{{champion:18}} in the bot lane.. mid lane phase i died to a sona{{champion:37}} q{{summoner:52}} and then a lucian{{champion:236}} q-auto{{item:3133}} which did like 1200 dmg total at level 9 or so. I had such a low amount of health! and I checked and trist had 1700 hp and i only had 1100. Later on I died to an ahri's{{champion:103}} q w combo and like 1 auto attack. I had support items like twin shadows{{item:3905}} and ardent censor{{item:3504}} rather than any life items. So my question is... am I not allowed to play these supports without building a life item{{item:3067}} early, every single time? The point of a support character is to NOT fight and NOT be able to do any damage, but simply allow for other people do gain opportunities to do more and take less damage. But does that justify them being able to die in 3 hits? So I bought a spirit visage{{item:3065}} so sona/ahri couldnt 2 shot me anymore and then we both had ~1750{{item:1028}}. and everything was fine but for a minute there I thought I might be living in a hellish world of always getting 2 SHOT as a support cc mage main. I'm used to playing janna{{champion:40}} and never even having to deal with anyone getting close but Even then sivir{{champion:15}} and I{{champion:40}} both got 3 shot by a jax{{champion:24}} who all in dove us... and ... apparently he can avoid being knocked up by my tornado now if he is spinning his staff... or maybe that was just some weird one time bug, but he literally did a leap strike right into it near its origin point (it was behind/on top of the sivir which he was mauling) I can see that a lot of the balance is in the fact that CC spells are often splash abilities which can end an entire enemy team in 1 move, also most supports have mobility and heal/shield abilities so their counterplay when engaged on is to stun and run. But... Do they really need 3/5 the base hp of an adc? Hey maybe its the ADC's who have too much hp lol.
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