As pointed out above the set up is completely different.
UK, Ireland and France have more racecourses, proportionally by geographical size, than the US.
For example, Kentucky and Ireland are about the same size, they have similar sized populations. Both are famously TB racing and breeding strongholds. Both have the same amount of racing days per year.
Kentucky has 5 race tracks.
Ireland has 26 (and that’s not counting Point to Point courses).
With the exception of Kentucky Downs (which is ironically very much in the UK/Ireland mold as a racecourse… few dates, none or little on course training), horses are stabled at train daily at the KY racetracks. Each track races for 4-5 days a week for months on end, and horses work on the track every morning 7 days a week.
In Ireland each racecourse holds only a day or two of racing, then sits idle for weeks or months before racing again. The longest meet in Ireland is the Galway summer meet in August, which is 5 consecutive days of racing, and the course is often pretty beat up come Saturday. The horses are all trained at home on farms or on public gallops at the Curragh. They only go to the track to race, not train (there are occasional exceptions to this, when horses, usually high profile horses, will get permission to work on the course after the last race, or between races).
Races are only cancelled in Europe if the conditions are considered dangerous, and that usually involves snow or frozen ground in the winter. The will race on soft or heavy ground if need be. In the US by contrast, turf races will be put on the maintrack if their is even a small amount of rain, to prevent the turf course for getting cut up. Exceptions are generally made for G1 races, but even those can sometimes get pulled off the turf and onto the main course (the Shadwell Mile at Keeneland in 2013), though they often get downgraded to a G2 or G3 when that happens.
Dirt, being dirt, can’t really get damaged in the same way by rain. You just let it dry out and drag it and you are back to normal the next day, or even later the same day with enough sun and wind.