If you aren't happy with the default filtering on mobiles, you need to either change network to one you think can do this better, or pay for a third party solution, such as Kaspersky Safe Kids, which is a decent choice and as such solutions go, good value. For younger children this is a sound approach.
Bear in mind that as kids turn into teenagers and then into older teenagers, controlling and monitoring approaches become progressively less appropriate, and may not do any long term good or even be counterproductive. With or without filtering, you need to have age-appropriate discussions with them about what is acceptable and what is not, and as they go through teenage years you need to relax some of those rules, difficult though all of that may be. Having a discussion with a mid-teen son about what sort of things they should and should not look at is cringeingly uncomfortable for all concerned - but if you're not providing guidance that reflects their development into young adults, and relying on software that says "we're in charge, it's all evil", then where will they get guidance?
