How to implement searching and filtering in Ionic

You are doing it wrong.

1. You do return this.data.filter(...). It has no effect to your origin array (this.data). It just return a new array and does not make change to this.data. See the filter function doc
2. If you want to make change to your data you need to add: this.data = this.data.filter(...). But if you do like that, you will fall in other mistake. Your this.data will lost some element after filter and it can not be revert when you reset the filter.
   

So you should do like that:
In your component:

allData = []; //Store all data from provider
filterData = [];//Store filtered data

ionViewDidEnter(){
  this.allData = this.locations.data;
  this.filterData = this.allData;
}

setFilteredLocations(){
    this.filterData = this.allData.filter((location) => {
      return location.title.toLowerCase().indexOf(this.searchTerm.toLowerCase()) > -1;
    });
}

And in your template use filterData in ngFor:

<button ion-item *ngFor="let location of filterData">

Yo didn't send the $event here. So try as shown below.

.html

 <ion-searchbar [(ngModel)]="searchTerm" 
(ionInput)="setFilteredLocations($event)"></ion-searchbar>

.ts

 setFilteredLocations(ev: any){
    let val = ev.target.value;

    if (val && val.trim() !== '') {
      return this.locations.filterLocations(val);
    }

  }

You can see the official sample code here.