I'm not sure what the strong arguments in favor of conscious Turing machines are, but I'm aware of at least two reasons people deny it! ...the technical limitation of simulating (specifically) the human brain, and the mind-body problem. The mind-body problem has all the appropriate literature, so I will not attempt to repeat this, except to say that whatever a Turing machine could do, is done by "body".
In my capacity as a mechanical engineer, I have from time to time used commercial software that implements computation to simulate physical systems, such as finite element analysis of the stress at each element of a bracket or axle (etc) resulting from applied forces. The computer writes equations at each element with the goal of solving for an equality and uses iterative methods to arrive at a solution, since empirical solutions (as with differential equation solutions) are so rarely practical. Even so, the solution is fraught with interpretation, as when an internal 90-degree corner has a singularity in stress, the mathematical result of "stress concentrations."
If memory serves, a typical analysis that I performed a few years ago involved about 200,000 elements and my 2005-era computer could finish it in less than an hour. Using Moore's law and guessing that a transistor increases computation speed linearly, a computer today could do the same thing in between 1 second and 60 seconds.
Increasing the number of elements (or "degrees of freedom") is desirable because it increases the precision of the output, but at an exponentially increasing computer cost, as demonstrated here.
What would be necessary to simulate the human brain? The human brain has 85-86 billion neurons. If we take "Computational Time for Forward Elimination Steps of Naive Gaussian Elimination on a Square Matrix" as a reference, reasonably there is about an increase by 10 ^ 10 in computation time for each ten-fold increase in the number of elements in our computation, so when we add six factors of 10 to my 200,000-element computation, we need sixty factors of ten for the computation time.
However, my physical system computes elements that are only related to four or six nearby elements. Each neuron has on average approximately 7,000 synaptic connections. At a minimum, this can be multiplied by my element count for this purpose (of guesstimating a reasonable minimum computation time to simulate brain activity) so we need another thirty factors of ten on top of our computation.
My computation was linear, in that a given input in force causes a proportional response in stress, deflection, etc. Neurons are non-linear, so we must somehow account for this increase in computation time due to complexity. My computation was static, meaning that time-dependent aspects of the computation weren't part of the equation. Neuron activity is highly time-dependent (see this again).
Without addressing non-linearity and time-dependent issues, a brain computation would reasonably take a computer today between 1 and 60 seconds multiplied by ten to the ninetieth power. To reduce the computation time back to under a minute, Moore's law would need to work for the next 660 years.
My computation was for a single input and a single output. The human brain is constantly receiving inputs and producing outputs at an unknown or (more likely) stochastic rate. If this were 100 per second, then that tacks easily ten more years on our Moore's law estimate, placing a working brain simulation nicely in the year 2780.
Nobody believes that Moore's law will actually continue working for the next 670 years. Will quantum computers save the day? Google's quantum computer is 100,000,000 times faster. But this only shaves 8 factors of ten off of our 90, resulting in only 82 factors of ten for the brain computation and 550 years for Moore's law to work: computer brains by 2570.