/user_directory/search searches by checking the start of the string instead of the contents

This issue has been migrated from [#13807](https://github.com/matrix-org/synapse/issues/13807).

---

### Description

Using `search_term` `al` will return users that are named `alex` but using `search_term` `lex` won't return anything.

### Steps to reproduce

Execute a post to `user_directory/search` with `search_term` that contains the middle of a name of a user.

### Homeserver

Local running Synapse

### Synapse Version

{"server_version":"1.62.0","python_version":"3.9.13"}⏎

### Installation Method

Docker (matrixdotorg/synapse)

### Platform

MacOS 12.5.1 - Apple M1 Pro

### Relevant log output

```shell
Not applicable
```

### Anything else that would be useful to know?

_No response_