I wouldn't recommend migrating to Azure as the first thing to consider when trying to solve a performance problem.
You should try to understand why your queries are running slower. There is a wealth of tools available within SQL server to help you do that. For example:
- You can set statistics on - for instance setting time and IO can tell you the time taken and the number of reads for each statement within your query
- You can examine the query plan aka execution plan to see how SQL Server is implementing your query and what parts of it are performance bottle necks.
- If you are on SQL 2016 or later, you can use query store to track a variety of information about your queries over a period of time - e.g. regressed queries, queries that consume most resources, etc.
To use any of the above (and there are others), you will need to do some research. But the great thing about SQL Server is that there are plenty of examples and guidance available online. And, there are plenty of people willing to volunteer their time and efforts to help you get going.