Traveling with my bicycle on top of my car in Europe

In UK, this page says

Bridges that have a clearance of less than 16 feet and 6 inches, or around 5 metres will usually have signs.

There are often advance warnings posted, with an alternative route suggestion.

Low bridges are notoriously easy to overlook. The Cambridge News has

Another van has smashed into the infamous Ely railway bridge

You also have to watch out for height restrictions at car park entrances. This page says

• the actual height can be lower than the marked height

• the exit barrier can be lower than the entrance barrier

One of the down-sides of having a bike on the roof is more susceptibility to side winds.

Regarding Italy, this page has some links to the relevant legislation. Google translator does a good job converting the page to English.

The main points are:

• the bicycle carrier/rack must have the CE marking
• total height must not exceed 4 meters
• total weight must not exceed the maximum roof load as stated by the car manufacturer

Underpasses are clearly marked, in meters. In my experience (mainly driving in Northern Italy, mostly near cities - the countryside might be different) detours are readily available.

You're right to be careful, I've had "encounters" with low overhead entries on at my local supermarket and a park and ride in Oxford.

The difficulty with the supermarket is that I go there all the time, and normally don't have the bikes on top. When it was suggested at the last moment that we pick up an easy dinner after a tiring ride, I didn't re-evaluate my planned stops.

With the park and ride, I'd already done the 50 mile ride and was being collected from a spot chosen for clearance. Then on the way out, we changed our minds and thought we'd go into town for dinner -- but the parking lot we then needed to use had a barrier.

I've been lucky in the damage area, but perhaps a little model bicycle on the dashboard for when the bikes are overhead might be a good idea. :-)

On minor roads, there may be occasional super-low bridges and such, but they should be clearly marked.

I have found a roof carrier very convenient when driving. No significant added noise, bike stays clean even in bad weather, fuel consumption slightly higher but nothing major. In fact, the biggest risk may be forgetting that the bike is there and driving into a parking hall. You will probably destroy the car and/or the bike.

When driving on highways, cities, tunnels, bridges, under bridges etc. is there any case when an eventual high limit could restrict me in this case?

In France, some tunnels, bridges, toll lanes and parkings have a height limit that you may reach with your setup, but it's typically clearly indicated beforehand.

What are the downsites of traveling with the bicycle like that? What should I be careful with?

Slight increase in gas consumption due to lowered aerodynamics. Unlike what another answer claims, one doesn't have to pay more for the tolls on the highways, but sometimes mistakes happen.

In Sweden heights lower than 4.5m are marked with this sign, I couldn't find any relevant regulations besides the car maker's own limitations on weight

If you consider the trip using a passenger car typically sedan, estate/kombi or hatchback there is a no problem for you - the total height will be lower than big trucks or tall lorries. All bridges, tunnels etc. should be labelled with proper roadsigns.

If you consider using MPV, SUV or van, then there might be a problem, because these are sligtly taller. You shall still be okay with that because the trucks are really tall - the small roads may have some risks but very little.

On the other hand, if we consider fuel costs and tolls of the round-Europe trip, it might be a good time to count costs and benefits of mounting options.

• On-roof rack:
  pros - you have it, it does not obstruct entrances
  cons - huge drag, fuel consumption
• On-door mount:
  pros - inexpensive, lower drag
  cons - car-specific design, possible damage to the car (scratches and hits), obstruct the trunk door.
• On-jack mount:
  pros - easy to mount/unmount, low drag cons - expensive, obstructs the trunk door, needs the trailer jack.

The calculations may, but may not, end up that the on-jack mounted rack will pay off its price only by reducing the drag. And the faster you go, the more significant the drag is.