I agree that Messegelände needs a completely different translation.
First, 'grounds' is plural in this sense. However, it's still not the right word, because 'fairgrounds' (one word, not two) normally means an area where agricultural shows are held. These are annual fairs where people bring animals, vegetables, crafts, and baked goods, to compete for prizes. The fairgrounds are often an outdoor area with no buildings except large barns or sheds, though state fairgrounds have more permanent exhibition buildings. The word 'fair' alone in the US most often means this kind of fair: a county fair or state fair.
What CM2DD calls a 'fun fair' (BE), with rides and games, for us is a 'carnival' (AE), which is usually also part of a state or county fair, but not the main reason for its existence.
The word 'fair' can also mean a money-raising bazaar, like a church or school fair or a Christmas fair, with small booths with crafts and food for sale and games and contests for entertainment.
So basically, the word 'fair' means more or less Markt, Jahrmarkt, Volksfest, Kirmes, that kind of thing. (I don't know which German word is best for the agricultural show alone.)
But the word 'fair' alone does not normally mean Messe. That word should be translated 'trade show' or (less commonly) 'trade fair.'
The permanent site in a large city where many trade shows are held is sometimes called an 'exhibition center' if it's mainly used for events such as trade shows. However, in AE it's more likely to be called the 'convention center' because it's more often used for events such as regional or national meetings of professionals in a certain field.
If you are talking about a trade show from the point of view of your company, which is going to have a booth or exhibit at it, then you can say 'trade show/fair site/grounds,' as in, 'Be sure to take plenty of posters with the company logo on them to the trade show site / the trade fair grounds / etc.' This would mean any location where a trade show is held, not just the specific convention or exhibition center in a certain city.
In this context, Messegelände could be '[trade fair] [grounds]' (as separate words), but it should not be 'trade fairgrounds' (which would mean, if anything, 'to exchange fairgrounds,' as a verb). So basically I disagree with OALD sense 3 above. AE and BE don't really seem very far apart on this.