Is there any way to speed up city production?

A few options:

1. The most obvious solution is to increase the production of the city. A few ways to do it are:
   • Build a few production-increasing buildings before you build your long building.
   • Purchase production-increasing buildings with gold.
   • Increase the production yield of the city's tiles by improving them with workers, or build factories using great engineers (though great engineers can be used in a different way, see below). Marble is especially useful, as any city which has a worker on improved marble gets a 25% production boost, though only for wonders.
   • Shoot for technologies or social policies that increase production. For example Steam Power increases production from lumber mills, so if you have a lot of those around your city this technology could really give it a boost; Railroad will allow you to connect cities by railroad which increases production; and the Order social policy will increase all production for building by 25%.
   • Redirect the civilian allocation to increase production. You can just use the radio button under the "civilian allocation focus" to let the computer choose the best allocation, or you can do it yourself manually. It means that you should put civilians inside tiles / buildings that have the most impact on production. If you're desperate for production you can de-allocate a citizen altogether: each unemployed citizen produces 1 production per turn.
2. Chop down trees around the city. Chopping down trees gives a small boost towards the currently in-progress building. You can even chop trees in an area outside your territory, it still gives some production (though less, and it gets further decreased with distance) to your closest city.
3. The great engineer can speed up buildings and wonders. When you use his ability it gives a tremendous boost to the current in-progress structure, often making it complete just one turn after you use his ability. Very powerful, don't waste it on anything which isn't worth it. Notice that if you build something ridiculously difficult it will probably not get down to 1 turn.

Note: if it's particularly the courthouse which is giving you trouble, consider getting the Police State social policy, under the Autocracy tree, which makes occupied cities produce only half the unhappiness value.

Oak has a great list of the conventional methods for increasing production. There are a couple of other options you can consider, although they are more situational and involved:

Start a Golden Age

In a golden age, any tile that produces gold produces an additional gold, and any tile which produces production produces an additional production.

I love production I was producing producing additional production.

Golden Ages can be started using a great person, saving up enough excess happiness, and various wonders/social policies.

Play as a Civilization that Boosts Production

This is more involved than starting a Golden Age, or any of Oak's suggestion: Civilizations are chosen before the game, so you can't react in the middle of the game when you realize you need a courthouse to finish faster, and you lose the benefits of your other favorite civilization. So, I would recommend using other suggestions first, and switching Civilizations only if you find the other suggestions aren't enough.

• Rome: "+25% Production bonus when constructing a building (in another city) that already has been constructed in the Capital."
• Russia: "[Strategic] Resources provide +1 Production"
• Egypt: "+20% Production speed of Wonders."

Quotes from the Civilization 5 Manual.

Place cities in areas with plenty of hills and forest tiles. In Civilization IV, food tiles were of utmost importance in allowing cities to become large and prosperous. The introduction of maritime city-states to Civilization V, along with the decreased food output from non-strategic resources, makes food tiles a lesser concern. Now I happily settle in the hills and forests, knowing that in the mid to late game I can maintain alliances with several maritime city states to make up for any missing food. Besides allowing you to increase production by working mines and lumber mills, this also allows you to use specialists very liberally, which can lead to tremendous science output (with the Secularism social policy) and even more greatly increased production (if you are able to build the Statue of Liberty).

If you have a great engineer, you can expend them to complete a wonder instantly. There doesn't seem to be a way to increase production speed by spending gold, short of spending enough gold to purchase it outright (and not all units / buildings can be purchased. For example, wonders.)

Edit: As of the 12/15 patch, you can buy courthouses outright in conquered cities, which is a decent way to boost production in those places. (Though expensive!)