I was standing in the check-in queue, sunburned, exhausted and very late for a seven-hour flight from Toronto to London. My wife had, sensibly, returned from the holiday a day earlier. In one arm I held my screaming one-year-old daughter, Lyra. Her folded pram was slung over the other shoulder. I was clutching passports and a nappy bag and surrounding me on the floor were my suitcases and an assortment of carrier bags overflowing with food, nappies, books and toys. Lyra went silent, I felt her abdomen tense, her face turned deep crimson and she forced out a long and resonant fart which sounded far more like it came from me than from her. A little area cleared around me in the airport. For some reason, probably fatigue, I put a finger into the nappy gusset to check that it was just a fart. It wasn’t. Now my finger was coated in baby poo and pretty soon so were our passports, luggage, Lyra and me. She was unfed, unwatered, covered in shit, and we were about to miss our flight.

In truth, I knew I would be this kind of parent, even if other people had higher hopes. There is a widespread idea that being a doctor must give you a leg up in the parenting game. My wife, a journalist, is often told: “It must be nice to have a doctor as a husband.”

There are superficial similarities between being a doctor – especially an infectious diseases doctor like me – and the first year of parenthood. Faeces is the common currency of both tasks. Viruses abound. A friend who is a paediatrician likes to say that, up until the age of five, children are functionally HIV positive. We’ve had stretches of months where coughs, snot and bouts of diarrhoea have taken turns wrecking sleep for everyone while I remain helpless. Doctors, even infection specialists, don’t have a monopoly on treatment of minor infections. We know very little about what causes them and less about how to fix them. I have a PhD in virology but my mother’s ginger-lemon-honey recipe is the staple antiviral in our house.

So far there have been exactly two occasions where I felt skilful as a parent because I am a doctor. First, Lyra was born in a heatwave. I kept her hydrated and avoided admission to hospital for IV fluids even though we were struggling with breastfeeding.

The second display of skill came when Dinah had an episode of mastitis, a blocked infected milk duct. We used hands, expression pumps and, of course, a baby, to drain the infected breast. The best moments of parenting are inextricably mingled with the worst. All three of us naked in the bath at 2am, covered in rapeseed oil for the boob massage, the water cloudy with milk, and the fever finally subsiding – a high and low point in equal measure.

Of course, simply because I felt skilful doesn’t mean that I was skilful. I did nothing that any parent with a decent amount of common sense and the internet couldn’t have managed. But both occasions have something significant in common: I felt skilful because I avoided medical treatment. Of course this isn’t necessarily skilful; I might just have been lucky that nothing went wrong.

This is important to say because it’s likely that doctors under-treat their children, likely because we ourselves avoid medical treatment. There aren’t masses of data but we know that what doctors want for themselves at the end of life and what we recommend to patients is quite different. We don’t want to be resuscitated or have dialysis or chemotherapy or feeding tubes. We know that in terms of mental health and suicide we are sicker than average, yet we present late for treatment, often only when compelled to by colleagues or the law.

It’s not altogether clear why we reject our own methods. One study of medical patients in the BMJ divides clinicians into perfectionists, narcissists, compulsives, denigrators of vulnerability, and martyrs, none of whom are likely to seek care. A few of us combine these personality traits to make someone who would literally die before seeking the opinion of a colleague.

With our children, we may be more likely to miss a broken leg or pneumonia than to turn up at A&E with a false alarm. I know that because I have two close friends, both excellent doctors, who missed those very diagnoses. It was their non-medical partners who insisted on going to hospital.

Even if I have been skilful twice, there have been many more times when I failed Lyra as her physician. Discoid eczema confidently diagnosed as a fungal infection. (Did I mention that I specialise in infectious diseases? When you have a hammer, everything looks like a nail.) Diarrhoea that I realised one day had been going on for several weeks without any attempt at diagnosis, all the while Lyra losing a dangerous amount of weight. This fact was only discovered by our excellent GP. We had never weighed her.

So, the idea that a doctor would be a better parent is bizarre in many ways. Perhaps the expectation that they would be speaks to the extent that we have medicalised childhood. Or maybe we think skills are more transferable than they really are. Do people who are skilled at something else make better parents? Would you expect Garry Kasparov or Tim Peake or Judi Dench to have “better” children? If you’ve ever met the children of doctors you’ll know that they’re no better adjusted, healthier or less injured than the rest of us.

But if being a doctor doesn’t seem to have improved my parenting, being a parent does seem to have improved my doctoring.

In the early months, the doctor in me approached my daughter as a set of problems to be cleaned, fed, silenced. I thought parenting was a form of work, undertaken to produce a particular outcome. This type of approach has parallels with much of the problem-oriented, metric-driven way that we practise medicine now.

But my wife, and in fact most of the people I know, seemed to have a different idea about parenting. I remember asking a friend why he didn’t do more regimented sleep training with his kids, who kept him up. “Because it’s not very nice for them,” he replied. This was a revelation to me.

Professor Alison Gopnik, a developmental psychologist at University of California, Berkeley, uses the analogies of carpentry or gardening to describe different forms of parenting. Carpentry is about the attainment of perfection. Gardening is about creating an environment where life thrives. Now we’re all about gardening.

The gardening approach may sound laidback – “I just let her be what she wants to be” – but it still requires a few technical skills. Part of being relaxed is knowing what to obsess about. When I plan, pack carefully, and get the basics of food, water and sleep right we’re all much happier and cleaner.

Mainly, I’m lucky that my wife is that rare person who combines being fun with being ferociously well organised, so that check-in scenario doesn’t happen too much these days.

I’m almost ashamed to admit the way that being a parent has changed me as a doctor. As I’ve started to see parenting, and Lyra, as things to be enjoyed, I’ve reflected on my approach to a busy clinic or ward round where patients can become units to be processed, problems to be solved.

My wife consciously taught me that our daughter is a small human being to be lived with and cared for. When she cries she’s not being manipulative; when she craps in an airport queue it is not to annoy me. Nappies aren’t things to be forced on and off as quickly as possible. Hands should be warmed before applying eczema cream.

I wasn’t cruel to her but I think I treated her like a patient, trying to solve her problems as efficiently as possible. Many of my patients are unwell and unhappy because of problems I can’t solve: the structural violence of their life and their environment. I can only listen and show that I care. So now I try a little harder to treat my patients, like my daughter, as human beings. I wash my hands frequently with both.

Chris van Tulleken appears in Operation Ouch! Live on Stage, at the Apollo Theatre, London, from 6 December to 6 January (operationouchliveonstage.com)