The full answer to your question would be a book on the environmental and demographic history of Mesopotamia, so I'll go for the short version. Mesopotamian civilisation originated in the Mid-Holocene global climatic optimum, the warmest and wettest period of the last 100,000 years. This also corresponds to the time when sea levels reached their peak: the Persian Gulf extended much further into Mesopotamia than it does now, bringing with it maritime air currents that significantly decreased the aridity of the regional climate. The result was that, in the time of the Sumerians, Mesopotamia was more heavily vegetated, the desert had retreated as far as it was going to go, the seasonal floods of the Tigris–Euphrates were at their most dramatic, and there was more water in the ground for irrigation. It was probably a big part of why cities evolved when and where they did.

After the optimum, global climates became gradually cooler and more arid again, and the upper reaches of the Persian Gulf began to fill in with sediment. Mesopotamia is therefore significantly more arid today than it was 5,000 years ago, purely because of natural climatic and geological processes. Against this backdrop, it's very difficult to say conclusively what effects humans may have had in exacerbating (or ameliorating?) the natural trend towards aridification and loss of vegetation.

What we can say for certain is that prevailing narrative—the one you mentioned—of Mesopotamia and Southwest Asia in general being an man-made, ruined landscape has been heavily conditioned by Eurocentric biases. When British and French colonialists began extending their tendrils into the region at the start of the 19th century, they really didn't understand arid environments. Conditioned to view forests, tilled fields and green pasture as the mark of a fertile and "well managed" landscape, they looked at drylands and saw a wasted, worthless landscape: a desert. Europeans' fetishisation of the ancient history of the Near East (especially its biblical history) was informed by, and in turn reinforced, this prejudice. When they saw a magnificent ruin they thought to themselves, this couldn't possibly have been built in a desert. Ergo, in ancient times Mesopotamia must have been lush and green, like Europe! And if it was like that then, it could be now: the presence of ruins proved that something terrible had happened to the land to put it in its present state. It suited colonialists to blame this on "the Arabs" (particularly nomadic Bedouin), because in doing so they were able to conceptually separate the modern inhabitants of the region, who they had little respect for, from their ancient heritage, which they idolised and coveted; and simultaneously to justify European "stewardship" of the region on the grounds that the natives had made a mess of it.

In the 19th century this prejudice took hilariously crude forms, like planting trees in the desert to "bring the rain". But it continued to profoundly shape Orientalist perceptions of Southwest Asia into the 20th century too. After Victorian-style naked colonialism ended, it co-opted the burgeoning environmentalist movement, taking the form of UN-backed panics about overexplotation, drought, and "desertification" in arid zones, as well as a triumphal zionist narratives of the "greening of the desert". Archaeology was again drawn into a circular chain of logic. Archaeologists accepted that the landscape they saw in front of them was "degraded", and therefore neglected to explore how people might have, for example, built cities and civilisations in an arid landscape. Instead they looked for explanations of why things had changed so much, and invoked the things that environmentalists were talking about: overgrazing, salinisation, deforestation, etc. What the archaeologists were saying about ancient environments then in turn bolstered the narrative that places like Mesopotamia had been turned into a desert by irresponsible land use.

I don't mean to undermine the scientific work that went into these explanations, but the whole thing was a naked emperor situation. The "desertification" narrative has largely collapsed in the last decade becayse ecologists have come to realise that there just isn't enough evidence it actually happens. Instead they've come to appreciate that arid lands are highly variable, but resilient, systems that might go through long periods of drought, and then bounce back, perfectly naturally.

Archaeologists have been slower at realising the emperor has no clothes on. But in my opinion the upshot of this new understanding is that while anthropogenic environmental change always remains a possibility, we should be much less quick to invoke factors like salinisation, over-grazing or deforestation, especially when we know there has been such marked natural environmental change. People can and do badly mismanage environmental resources in drylands, but there's little solid evidence that they do so in a way that can permanently scar a landscape for millennia.

(I wrote expanded on a lot of these points in answer to a similar question here last year: Why is the Fertile Crescent now desert?)