He doesn’t see I’m more tired

Cut yourself some slack. Your body has been through a hugely traumatic experience and is adjusting to less rest. It is my belief that after you give birth your sleep is never the same again. As a bonded mumma you are programmed to be on high alert for your little ones needs at all times and that's a good thing.
Your partner is probably exhausted too. It's a hard phase in life. If you are concerned that rest is not restoring your energy levels, maybe explain it to him as just that. "I'm worried that rest is not giving me energy and am not sure what to do about it?" Have you spoken to your doctor and had your iron levels checked? I would be starting there. You could also be a little post natal. To my knowledge deppression of any kind can be exhausting in itself. X

From experience op the whole whose more tired argument is pointless and will only erode the relationship away. You'll both be tired, yes you will be more tired but he's also likely to be the most tired he's ever been so he's unlikely to admit it's worse for you. It's bad for all involved. You need to work as a team instead to find a solution to a hard part of parenting. I really feel for you so please understand this message comes from a place of advice 😊

He’s allowed to also be tired, which I’m sure he is if he’s doing half the nights then working the next day. I totally get that you have health issues and he doesn’t but don’t go down that road because nothing good will come of turning into a competition. Instead of fighting with each other, remember you’re a team and focus on the practicalities of how to make sure you get more sleep. Regular childcare e.g. a mother’s help or babysitter is a great idea if you can afford it. You could also try a rockit for the pram if baby likes movement and have that going in the living room whilst you sleep on the sofa. Or safe cosleeping. Sleep training when the baby is old enough could also be an option.

As above, the 'who is more tired' competition will never be won and just breeds resentment. Give it up.

instead, channel thoughts into practical ways to help yourself. Working on getting baby to nap in cot/moses basket will give you space to nap in the day - often babies need some training or support to get them used to this, so work on that. Try different methods and see what sticks (you do need to try for more than one occassion of course!) - swaddling, white noise, RockIt rocker tool thing to give movement, gradual retreat, patting baby whilst they lie in bed, dummy for comfort etc.. It will be time well spent if the end result is that in a few weeks' time you manage to have a couple of free hours per day to nap whilst baby naps.

You absolutely need a mother's help if you can afford one. I had a 19 year old who was studying childcare at the local tech come in for parts of most days. Sometimes she had the baby, sometimes she did housework. Baby loved her and I got to rest when I needed to.

So you knew you had sleep issues before having a baby now you are acting like one by competing 'i am more tired than you'

If he works I'm not sure how he could help you with having naps in the day. I would say follow the advice for safely co sleeping in bed and then you can actually sleep when the baby sleeps. As long as you don't drink or smoke and the baby isn't premature then it can be safe - definitely safer than falling asleep on the sofa with them. I completely understand the struggle not being able to sleep if your baby likes contact napping, so this is the only way I can see that you might be able to do it

Competitive tiredness is pointless and isn’t good for anyone, tired is tired. Me being more tired than my husband doesn’t make him feel any less tired, it’s hard with a baby, my daughter is 8 months now and one of the things we agreed on from day 1 was never to complain to each other about being tired because it doesn’t help and the reality is we both are.

It’s easy to always think the other person has it better or easier but honestly that’s not helpful or even necessarily true. Focus on practical things, for example is it possible for you to have an hour to have a nap when he gets home and then you take over so he can have a nap? In the early days with our daughter when we were both home we would take it in turns to have a quick power nap in the afternoon/evening.

I am more tired than you is a petty and childish competition.

Who is more tired is the worst argument, it's pointless and divisive, and doesn't help the situation.
I would ask him to take the baby out for an hour or two as soon as he gets home so you can sleep
It doesn't last forever, but helping each other is way more effective than making someone say you have it worse than them

Can you afford nursery?

What illnesses do you have? Is treatment helping at all?

Sounds really hard going but please do try and avoid turning it into a competition. Doesn't matter what he feels about his own tiredness and yours. What do you actually want him to do? Focus on that.
You say he is home a lot and parents (I don't agree with "helping out"! Let's just call it what it is - he is being a parent). What more would you ideally like him to do for you? Does he need to work less? Is it financially feasible?