About every single team has a following, but define “strong”. Many minor league teams don’t have enough of a following to sell nameplated T-shirts, but so long as they have enough of one to maintain their field they’ll keep playing.
In Spain the Liga is divided into several divisions; the uppermost is Primera (First), with teams from all over the country including (most of) those who go to international competitions; Segunda (Second) is country-wide as well; Tercera (Third) is divided into 18 regional groups (which may not necessarily correspond to administrative regions). There’s even-lower groups below that, which are organized by the 19 regional federations and vary names by region.
All of them have enough of a following to pay for a playing field, and the majority (at least from Third on up) have a school which may rank from “slightly more prestigious than having your kid play for his school” to “OMG can I have your kid’s autograph?”
Primera has 20 teams. Every year, the worst ones move down to Second and are replaced by the best from Second (I think it’s 2 right now but that there have been times it was 3); similar promotions are held at every level. The most Primera has had was 22 teams, for two seasons in the 1950s. Four of the current Primera teams are owned by their fans (Athletic de Bilbao, Barcelona, Real Madrid and Osasuna), the rest are the property of Corporations which work like any other company; Bilbao, Barça and Madrid are also the only three teams which have always stayed in Primera. There’s a total of nine teams which have never been below Segunda. The oldest team in Spain, Recreativo de Huelva, is not currently in Primera. A team may get kicked down to Segunda if it fails to meet its financial obligations; this could (and has) save another team which didn’t play so well but did pay.
My brother follows both Barça and Osasuna, and for the longest time couldn’t even say for sure which one was “his” main team. He eventually figured it out one day that our Barça-Barça-Barça cousin dissed Osasuna… yep, Osasuna first. Like him, many people follow one or more local teams as well as one of the big ones.
There is a yearly competition, Copa, which involves every team from Primera and Segunda. It raises as much passion as the Liga itself, specially when a Segunda team manages to kick out a big Primera (specially Madrid or Barça; people love kicking a fallen giant). At the same time, when international competitions come up, people will forget local relationships to cheer for “our” team… mind you, this has been known to cause trouble deciding which one is “our” team when we had, say, a Manchester full of Spaniards and coached by a Spaniard meeting a Barça where even the dog spoke foreign.
The Copa winner gets to go to international competitions, so you can have a Segunda travelling abroad to meet some of the biggest names in the sport.
Have I confused you enough?