I can think of two far better ways to balance districts. The first makes more sense to the way Civ usually works. The second would be a departure from the norm, but still work pretty well.
1. With a base of 60, scale each copy of the district with the greatest number by 10, the 2nd greatest by 20, etc.. So, for example, you build a Campus and Holy Site in your first city, and they both cost 60. In your second city, you build another Holy Site for 70. After that, another Campus would cost 80, or any of the districts you haven't built would cost 60. This method encourages either specializing one victory condition for your empire, or specializing each city to its greatest benefit, but discourages trying to build all districts in all cities. So if you had 8 towns, and you built each separate specialty district in the 8 towns, it would cost 480 production total. Quite easy, and each city is specialized, but not overly powerful. Focusing on one particular district for a victory condition in 8 cities would cost 760 production. Also pretty cheap, but you wouldn't be very well-rounded. Putting two districts in each city would cost 1800 production, possible by the middle of the game (the final city would cost 330 production). But the costs would quickly balloon, with the final city costing 3000 production to place 8 districts if all 8 cities had 8 districts. The scaling could be adjusted up or down a bit, but this method worked very well for a MUD that I played long ago where you could multiclass into 5 different classes. Maxing out the first class took some time, but was fairly easy. Maxing out all 5 took a very long time. You could mix and match pretty easily for a while, and even switch focus with some work, but the scaling never got insane unless you really wanted to max out everything.
2. Districts could be built regardless of production. There are a couple ways to do this. You could either make it, say, a flat 5 turns for the first district in any city, 6 for the second, 7 for the third, and so on. Or you could make them similar to Great People/border growth, and remove them from the build list altogether. Start with 50 points needed (add 50-100? each successive time), and add the population of the city to that counter each turn. That way it essentially scales with food/housing instead, which makes a lot of sense if you think about it. The buildings in the district would still cost production, but the districts themselves would be chosen once the meter is filled. It also gives a very good reason to have high culture or buy tiles for when it's done. This wouldn't change people choosing the 'best' district in the game in every city, but at least production wouldn't be so necessary (in fact, it would give high growth/low production cities a boost), and it wouldn't oddly handicap science/culture advancement.