Several parts to this.
The thoroughbred has been bred for competition for more than 350 years. Horses were raced in England from Roman times, and doubtless somewhat selectively bred since then, but the 1600s marks the period of importation of Eastern bloodstock (general called 'Arabs' but in reality more of them being Turkoman/Ahkal-Teke). The Eastern horses weren't as fast the domestic product, but positively altered the biomechanics when crossed with the).
With more than 350 years of selective breeding there has to become a point where incremental improvement is so small (if at all) that it's virtually imperceptable.
That point was probably reached in the 60s and 70s.
What has happened since is that while the top end isn't moving forward or is doing so impercetibly, the average is still progressing. So fairly ordinary horses are recording times which would have been noteworthy 60 years ago.
With running the trend has been for tracks and shoes to become faster, and training to increase in volume and intensity.
With horses tracks (if we are talking about North American dirt tracks, mainstream US racing) the trend overall has been to them being deeper and slower. A faster shoe isn't going to happen any time soon (aluminum plates were already commonplace in the 1930s).
Training is one area where improvement is possible. Very few trainers have much knowledge of exercise physiology. Increasing volume and intensity isn't likely to be effective (we're not going to see an equine Cam Levins).
What could happen is monitoring of horses and recovery rates could lead to more efficient training with regard to the windows in which a horses is working, and also a better management of when a horse peaks.
I'll finish by saying 'too much inbreeding' is something of a myth. To begin with there are multiple measures of inbreeding (at least four, some dealing with close inbreeding close up in the pedigree another, the Ancestral History Coefficient with deep inbreeding) and paper inbreeding (what the pedigree shows) and genetic inbreeding are not identical. There have been great horses with a wide range of inbreeding, from the European horse Coronation V, who was by a son of a stallion out of a daughter of the same stallion, to Secretariat, who was a complete outcross at five generations.
There are, however, optimal ranges for all of these coefficients, generally edging towards lower coefficents for inbreeding in the first five generations, and higher coefficients for deep ancestry (the pattern of Secretariat).
What most people are think of here is what's know as "Inbreeding Depression" - caused by increased homozygosity from partially recessive detrimental variants. There is some danger of the thoroughbred suffering from this across the board (as opposed to individual examples), but this point has what to occur. There are some negative alleles (for cartilage formation for example) that do occur more frequently in inbred horses, although that is tending to impact soundness (likelihood of being injured) rather than speed.